) ,
^2 = c2m2 â€” v2m2 (Z2 cos cf> + n2 sin ^>),
On writing down the equation of the tangent plane at Pt
we find that it meets the plane of xy in the line
/il + ^coscftx ^=1
and that the tangent plane at P2 meets the plane of xy in
the line
/ 12 Vo cos d>\
^ = 1
We thus have
ml cl {^2
m2 c2 fii
p>
where fil is the absolute index of refraction of the first medium
and fjL2 that of the second, fi being the index of refraction
from the first medium to the second.
150 On FresneVs Convection Coefficient.
We have, also,
c2 fX2
If 6] be the angle of incidence and 62 the angle oÂ£
refraction,
cos 0l = 7i1, and cos 02 = n2,
h 4-
Vl
cos
sin #i = V 1 12 + ?n12, sin <92 = Vtf + â„¢22 ;
v2 cos $ Vx cos (j>
also ml = fjLm2, and li â€” fd2 + /a
.-. Z12 + mi2 = At2[Z22 4- V] + Vzs(a*
snce
that is, if
6'2 Ci
1'2 COS (/> Vi COS <Â£s
Hence sin @i â€” a sin 0<>, if = â€”
r c2 cx
or if
fjb2 C\ _^1_/C1V__ K\
fXl C2 V2 fC2V K2 '
2*
K2 pi
Thus, if cx be the velocity of light in an}' medium at rest
in the sether, and if /^ be the absolute index of refraction of
the medium, the sine law of refraction will always be satisfied
if we assume that the velocity of light relative to the medium
when moving with velocity v in the sether, is
Av
Pi
where A is a constant which must be the same for all media.
Further, the condition that the incident ray, the refracted
ray, and the normal shall lie in the same plane is
li mx
2
Therefore
L m<
, l'! COS (f) l'! cos (f)
^ -| â€”
6*1 _ lx cl i\ c2 cx
k+'
V2 COS (f) l2 V2 COS
(
%
Fig. 1.
61 82 83 81 85 86 87
Tl Pb Bi ?o N*
88 89 90 91 92
Ra fie Th Bv U
Â£â€ž 5 I ^ I 5 I 6 I 7 j 0 I I I 2 I 5 [ 4 I 5 I 6
Â£k- â€” (Si)**â€” C^! >
w 4& m: ~~^ #
&A
vv
SÂ¥
weight) is the same for each isotope lying on the same
southwest to northeast axis ; the number of electrons in
the unstable next inner ring (E;) is the same for each
member on the same northwest to southeast axis ; the
number of valency electrons (Ew) are shown on the north to
south axis, and the total "metastasic" electrons (Em = Et-+ Eâ€ž)
are the same for each substance on the same east to west
axis. In the partial atom-structural formula the nucleus u
for uranium nucleus with atomic weight 206'4, and t for
thorium nucleus of atomic weight 208*2, contains the
unknown number of helium nuclei and electrons and
indicates respectively lead from uranium and lead from
thorium. To this nucleus are added the number of He, E*,
* Loc. cit.
Tlie Construction oj a Parabolic Trajectory. 157
and Eâ€ž which constitute then the partial atom-structural
formula. Thus e. g. the four isotopes of element with
atomic number 88 are as follows :
AcX=w . He4E8+2 with an atomic weight of Â«-i-4He = 222*4
ThX.= *.He4E8+2 â€ž â€ž â€ž *4-4He = 224'2
Ra=Â«Â« . He5E10+2 â€ž â€ž â€ž Â«+5He = 226-4
M3th=i . He5E10+2 â€ž â€ž â€ž *+5He=228-2
As 'has been pointed out previously * this partial atom-
structural formula indicates
(a) the series to which the isotope belongs by u resp. t ;
(b) the atomic weight, by the addition of the number
of He (4-00) to u resp. t ;
(c) the valency, by the number of valency electrons (Eâ€ž)
written at the end ;
(d) the number of metastasic electrons lEm), by simply
adding the inner and valency electrons together.
The relationship of this system of radioactive elements to
the periodic system is established by attaching figure 1 to
the lower part of the new periodic table tÂ« Whether or not
it is possible to extend this scheme of isotopes to the non-
radioactive elements is a problem of the future.
Berkeley, Cal.,
July 26th, 1919.
XI V . Note on the Construction of a Parabolic Trajectory and
a Property of the Parabola used by Archimedes. By
W. B. Morton and T. 0. TobinJ.
IF a particle is projected, in a given direction, from a
given point A with such a velocity that it hits a second
given point B, then any number of points on the path may
be obtained by the simple construction shown in fig. 1.
AH is drawn to any point on tho vertical through B, HI is
parallel to the direction of projection, then the vertical
IP meets AH in a point of the path. This construction
involves a simple property of the parabola which does not
appear in the ordinary text-books. If chords are drawn
* Phys. Rev. he. cit.
t Loc. cit.
X Communicated by the Authors.
158 Messrs. W. B. Morton and T. C. Tobin on the
from a point A on the curve to any other two points B, P,
and if the diameters through B, P are drawn to intersect AP,
AB respectively, then the line joining the points of inter-
section is parallel to the tangent at A.
Fig. 1.
Referring to fig. 2 in which the parabola is placed in the
more usual posture, assume that the points A, B and the
tangent at A are given, and construct P in the manner
indicated above. It can easily be shown that P lies on the
curve.
For AN : AK = PN : HK = PN : BM
and AK : AM = IK : BM = PN : BM
.-. AN : AM = PN2 : BM2.
It is interesting to notice that this property is really a
special case of Pascal's theorem about a hexagon inscribed
in a conic *. Let the angular points 1, 2, of the hexagon be
at A, 3 at B> 4, 5 at the point at infinity on the axis of the
parabola, and 6 at P. Then side 12 is the tangent at A,
23 is AB, 34 is BH, 45 is the line at infinity which touches
* This way of looking at the matter was pointed oat to us by
Mr. F. M. Saxelby.
Construction of a Parabolic Trajectory. 159
the parabola, 56 is PI and 61 is AP. The line of collinearity
of intersections of opposite sides is HI and the tangent at A
meets this at infinity.
To continue with the properties of the diagram fig. 2,
Fis. 2.
join HM meeting the diameter through P at W. Then
evidently XW = IP and BVVU is parallel to AP. Again
get the point V by joining HU. then VW = WX and
BV is parallel to the tangent at P. For, since NA = AT,
the figure BXWV is similar and parallel to PXAT.
From these results it follows that
PV : PI = WI : IP = MK : KA.
This brings us to a theorem used by Archimedes in the
course of his investigations of the positions of equilibrium of
a floating paraboloid of revolution, contained in the second
book of the work on Floating Bodies. In the sixth pro-
position of that book he proves that a paraboloid, the length
of whose axis has to the latus rectum a ratio lying between
3 15
the values ^ and -g-, if placed with a point on the circum-
ference of its base in the surface of the liquid and then
released, will turn, under the action of its weight and the
1 ()0 The Construction of a Parabolic Trajectory.
buoyancy oÂ£ the liquid, towards the position with axis
vertical. In the course of the proof he quotes, as known^
a property of the parabola which is an extension of that just
obtained. The source from which Archimedes derived it is
unknown.
Using the lettering of fig. 2, which agrees with that
adopted in- Heath's edition of Archimedes, the theorem in
its most general form is as follows : â€”
From a point B on a parabola ordinates BM, BV are
drawn to any two diameters AM, PV. Through any
point K of AM a line is drawn, parallel to the ordinate
BM, to meet PV in I. Then P V : PI = or > MK : KA.
Archimedes refers only to the special case where A is the
vertex of the parabola. A proof of this case, on somewhat
algebraical lines, is given by Dr. Heath. The alternative
geometrical proof now given exhibits, perhaps, in a clearer
light the connexion of the theorem with the fundamental
properties of the curve.
It has been shown above that the two ratios compared are
equal when I is the intersection of AB with the diameter
through P. It remains to prove the inequality in other
cases.
Let K'F be another position lying, say, to the left of
KI, and let it meet AH in W. Join MH' meeting UB in
W, PW in W", and BH in H". We want to compare
I'W" with IW. We have
WW" : BH" = AP : AH = AN : AK,
HH" : HH' = UM : UW' = AK : UW,
HH' :IP =AP:AN = UW:AN.
Compounding these ratios
ww" : ir = UW : UW',
WW" > IP and so r W" < IW.
If I/K/ is taken to the ri^-ht of IK it will be found in the
me ma
than IW.
same manner that WW"== TJ-f Lorentz italicized in the
quotation above. The condensation claimed by Planck's
modification of Stokes's theory, for the Sun as well as for the
Earth and for all other material bodies, is no longer devoid
â– of influence on observable phenomena. It suddenly acquires
physical life, so to speak.
* Notice tliat the aberration is a first order effect, while such
phenomena as that expected by Michelson-Morley are second order
effects (v'2/c2), so that the above condensation suiting the aberration up
to 1 per cent, will reduce the Michelson-Morley effect to one ten-
thousandth of its value, and thus practically annihilate it. There is thus
no need for making- o- larger than 10-2.
M2
164 Dr. L. Silberstein on the recent Eclipse Results
In other words, the discovery made at Brazil naturally
suggests the idea that the observed deflexion is due to the
condensation of the cether around the Sun*, and although one
has been an implacable enemy of any gether at all, for the
last fifteen years, one does not hesitate to point out this-
possibility â€” a last glimpse of hope, perhaps, for the banished
medium.
Let us imagine for the moment that Einstein had never
published his debatable, though undoubtedly beautiful, new
theory â€” not even that of 1905. Then it is almost certain
that the Eclipse result would readily be acclaimed as an
evidence of the condensation of the aether near the Sun, as
required by the theory of Stokes-Planck, and would encourage
the physicists to work out in detail the optical and associated
consequences of such a condensation. But even though
Einstein's theory has been published, and is being made
popular in a most sensational way, we cannot help clinging
to the said idea. I just learn from 'The Observatory' for
August that Mr. Jonckheere suggested some months ago that
refractions may, inter alias, he caused by "a hypothetical
condensation of ether near the Sun." My point, however^
is that such a source of refraction acquires a particular
interest if it is treated in connexion with the half- forgotten
theory of Stokes-Planck, when it ceases to be a detached
hypothesis.
It is in this sense and in such an organic connexion that I
should like to draw attention to this aspect of the subject.
Of course, the quantitative details of the suitable modifica-
tion of the optical, or the electromagnetic, properties of the
sether due to a radially symmetrical or any other condensation
have to be worked out carefully. It is not the purpose of
this Note to give a complete investigation of this kind, but
only some hints at its possibility. Such hints, together with
some remarks on the possible advantages of the advocated
theory, will occupy our attention in the following sections.
4. If, merely to fix the ideas, the Boyle law is still adhered
to, the condensation s = pjpoo outside a radially symmetrical
gravitating mass is given, as in (1), by
logS=^ (3)
If we assume, for places near the Earth's surface, not more
and not less than what is just needed for the theory of
* The logarithm of this condensation would amount, at the Sun's
surface, "by (1) a.nd (E), to the enormous figure = (p)=\d/' (5>
Then, in the state of equilibrium, and with dm written for
any mass-element in astronomical units,
~~
=f-> ....:. (6)
where r is the distance of the contemplated point from dm,
and the integral, representing the total gravitational poten-
tial, extends over all material bodies. <1> being a known
function of p, formula (6) gives the required relation. It
will be seen from the definition (5) that the dimensions
of (work per unit mass of aether) are those of a squared
velocity. In order to bring this into evidence, let us recall
that
â€¢=>/f â€¢â€¢â€¢â€¢â€¢â€¢ e>
is the velocity of propagation of longitudinal waves in any
compressible non-viscous fluid *. This velocity is, in general,
a function of p, and becomes a constant for the special case
of Boyle's law, namely, our previous 1/ v/Â«. Using (7) and
writing, as bef ore, â€” = d log s, we have
~~

~~ equal to about 2*5 km. per second *. This is quoted
by the way only. But the ratio of these two velocities will
be seen to acquire a particular interest in connexion with
the recent astronomical discovery.
6. Let c, as before, stand for the propagation velocity of
light in uncondensed aether, i.e. in absence of, or far away
from, gravitating masses, and let c be the light velocity at a
place where the aether has undergone a condensation s. The
question is : How are we to correlate c' with s ? In other
words : On what are we to base the optical behaviour of the
aether modified by a condensation ? The only reasonable
* If so. then the condensational disturbances' due to the Earth and
other planets, whose velocities exceed fc, will be confined to conical
regions as in Mach'a famous experiments.
168 Dr. L. Silberstein on the recent Eclipse Results
answer is: On experience. For, clearly, we cannot deduce
a relation, which is essentially electro-mechanical, from me-
chanical principles alone, or from electromagnetism alone.
Nor can we imitate the usual dispersion theory (which makes
use of both kinds of principles), for we are interested in
those portions of the aether in which there are no atoms and
no electrons.
In short, as was announced in section 3, let us write down
the required relation by utilizing the observational result
obtained by the Eclipse Expedition. In other words, let us
see what that relation must be like in order to give the
observed effect.
â– Now, if we disregard the small discrepancies (which may
he either due to accidental errors or, perhaps, due to a
superposed slight ordinary refraction), the observed total
deflexions of the rays passing near the Sun are represented
by Einstein's formula (quite apart from his theory)
r0 c2
where r0 is the minimum distance of the (undeflected) ray
from the Sun's centre, and it can easily be shown that such
will be the case* if the refractive index n = c/c' at an}^
distance r>R from the Sun's centre be determined by
n2 = 1 -+ -rr ,
cr
or, denoting the potential by 12, and generalizing to any
distribution of gravitational matter,
n>=l+f. ...... (9)
[This, in fact, is the formula which would follow at once
from Einstein's approximate line-element
20 90
ds* = W(l - ~) - (dx* + df + dz2) (1 + ==),
for a "static "field.]
In order to obtain the required relation, that is to say the
assumption to be made on the optical behaviour of the con-
densed aether, it is enough to combine equation (9) with
our last equation (8), which gives
if.
dlogS (10)
* Approximately, that is, for small Ad, and consequently for
refractive index but little differing from unity.
and Stoke s- Planck' s JEther. 169
Such, then, would be the required refractivity of the
-condensed aether, obeying any law p=f(p). In particular,
if it obeys Boyle's law, we have
n2= 1 + 4-2.logs, .... (10a)
which is of a surprisingly simple form, and reads: n2 â€” 1
â€¢equal to four time* the logarithm of condensation multiplied bg
the squared ratio of the two velocities of propagation charac-
terizing the cether.
Notwithstanding this temptingly simple form of the
relation, I shall not try to "deduce" it from things more
familiar. I prefer to regard it as an assumption, dictated by
observation.
If the reader so desires, he can write n2 â€” l = 4icjc2} where
iv is the work, per unit mass of matter, done by the gravita-
tional field in condensing the aether. The small fraction
n2 â€” 1 being known from the Eclipse results (for any r), the
numerical value of this work is determined without any
further assumptions. If we agree to the lowest estimate of
log s at the Sun's surface, as required by the aberration
theory, we can also evaluate separately the ratio v/c, as
already mentioned. This, however, is only a secondary
matter.
7. Some details and further implications of the Stokes-
Planck aether theory, supplemented by assumption (10),
must be postponed to a later opportunity. Here it will be
enough to add only a few more general remarks. It will be
kept in mind that the proposed theory would account not only
for the observed astronomical aberration and for the older
terrestrial optical nil-effects, but manifestly also for the nil-
effect of the Michelson-Morley experiment. The bending of
rays round the more massive celestial bodies would be only a
by-product of the theory. Again, in view of the exceedingly
small condensation of the aether round single atoms or cor-
puscles there will be no difficulty in working out a satisfactory
â€¢electromagnetic theory of ponderable media. The proposed
theory would also have the advantage of not predicting the
obstinately absent gravitational shift of the spectrum lines.
It might also react, in part at least, upon the 1905 relativity,
depriving it of its indispensability in most cases, but by no
means banishing it from the whole domain of physico-
mathematical investigations. Finally, the just objections
raised by the advocates of the physical principle of causality
ugainst the fixed and homogeneous aether of Fresnel-Lorentz
would not apply to Stokes's modified aether. For this
170 Sir Oliver Lodge on a Possible
latter would by no means be a mere framework of reference
axes and, as such, illegitimately privileged. For in referring
a class of phenomena to the sether here advocated we would
ultimately refer them to assignable physical things, namely
those most massive gigantic bodies which, so to speak, have
the strongest grip upon that medium. It is, among other
things, this latter -remark that I hope to make particularly
clear at an early opportunity.
London,
December 22, 1919.
XVI. Xote on a Possible Structure for the Ether.
By Sir Oliver Lodge*.
DR. SILBERSTEIN'S communication gives me an
opportunity for calling attention to a paper of mine on
many points in connexion with the ether which mast surely
be of interest even to those who are contemplating the
abandonment of that medium. In that paper an estimate is
made of etheriai density, and an attempt to measure experi-
mentally its lower limit is described ; there are also comments
of interest from Sir Joseph Larmor and Sir J. J. Thomson.
The paper is in the Phil. Mag. ser. 6, vol. xiii. pp. 188-5(jb',
and is of date April 11)07 ; though among other things it
relates experiments conducted in and about 1893.
The transmission of transverse vibrations like light shows
that the ether cannot be a mere structureless fluid ; and if it
is to be treated dynamically, which at first is surely a legiti-
mate attempt, it must have properties akin to what we call,
in matter, Rigidity and Inertia. Its inertia must be something
fundamental, which underlies and accounts for the inertia we
perceive in matter, possibly in a way having some analogy
with a motion of a solid through a perfect fluid. For when
an electric charge is moved, a magnetic field in the shape of
an ether vortex-rin^ is generated (with an energy of circula-
tion per unit volume equal to fju(eu sin By/Sirr*), and this
confers upon the charge its observed momentum if the
medium has the requisite density (see Phil. Mag., April
1907, vol. xiii. p. 492). The rigidity may be explicable
hydrodynamically by a vortex circulation â€” a turbulent
motion having a circulatory velocity of the same order as
that of the waves which the medium is able to transmit.
In Lord Kelvin's laminar vortex arrangement the velocity
* Communicated by the Author.
Structure for the Ether. 171
of wave-propagation comes out -J\/2 or "47 oÂ£ the average
velocity of turbulent motion (see Phil. Mag. for October
1887, p. 350). In all investigations the two velocities
come out of the same order: and in FitzGerald's collected
papers, No. 53 and No. 91, the two velocities can be identical
tor a certain arrangement of turbulence (cf. pp. 259 & 256).
On [age -457, FitzGerald expresses his tentative opinion that
the hypothesis that " the ether is a turbulent liquid has great
possibilities underlying it." And, again, on p. 486, "there
seems very little more besides interpretation of symbols to
make a turbulent liquid a satisfactory explanation of the
structure of the ether/'' Some assurance of stability may
also be needed.
Many things show that any granular structure which may
thus be possessed by the ether must be of a fineness incom-
parably minuter than any dimension associated with the
material units on which we can experiment. In fact, the
ether may quite well contain a linear dimension of the order
pQ-30 or X0~33 centim., and an energy of 1030 or 1033 ergs
per cubic centimetre (Phil. Mag., April 1907, p. 493). The
calm self-sufficient way in which it sustains all our stresses,
and transmits all our energies, shows that anything we can
impose upon ether is as far from perturbing it, or calling out
even second orders of small quantities, as the slight bias of
an ordinary draught of air is from perturbing the normal
motion of the molecules which compose it. A bullet in
air and an electron in ether can, however, attain perturbing
velocities ; and the fact is bound to be instructive when
increase of mass with speed is fully assimilated and its
mechanism understood. As said on p. 490 of the Phil. Mag.
for April 1907, retaining the meaning but slightly improving
the wording : The reason for the concentration of magnetic
intensity at the equator of an electron, moving with something
approaching the velocity of light, is that the flow associated
with and indeed constituting the magnetic field is then no
longer a small fraction of the intrinsic rotational velocity of
the ether itself (see also loc. cit. p. 494).
To explain gravitational and other facts, we must assume
that the very formation or existence of an electron sets up a
radial strain or tension all round it, varying as the inverse
distance, and likewise reduces the circulatory energy in its
immediate neighbourhood ; not necessarily causing any
change of density, since electrostatic facts (notably the
Cavendish experiment) show the ether to be practically, and
probably actually, incompressible, but affecting its elastic
or dielectric constant in such a way as to modify the velocity
172 Sir Oliver Lodge on a Possible
of light in the neighbourhood. (Compare Lord Kelvin,
Baltimore Lecture?, p. 465.) An electron might, in fact,
be a small region in which intrinsic circulation has ceased,
so that it possessed inertia only.
The tension or reduction of: pressure set up in the neigh-
bourhood of such a centre of force could explain gravitational
attraction, and a change of rioiditv would also suffice to
explain the very minute reduction in the velocity of light.
The refractive index needed at any point is 1 -f 2yMlrc2, or
l + u2/c'\ where u is the velocity of free fall from infinity,
which is just what the light has done. The dielectric
constant would be modified so that K/K^ = 1 + 4:jM/rc2 ;
the second term being Einstein's deflexion. It may be taken
as representing the deficiency of etherial circulation-energy
near a massive body, as compared with the unmodified
circulation-energy in free space. Just outside an electron
this deficiency is of the order 10~42 ; though just outside the
Sun it is of the order 10-5.
This note is hardly germane to Dr. Silberstein's paper ;
so I may just add that the complication of introducing
compressibility, and not only compressibility but an enormous
gravitational compression, in order to evade rotationality in
a hypothetical ether dragged by moving masses â€” for absence
of velocity-potential is well known to complicate unduly
the theory of astronomical aberration â€” does not commend
itself to me. An incompressible ether, not viscous at all,
is far more simple ; and astronomical aberration then
follows, as easily as on the corpuscular theory, without any
ingenuity.
But, as speculation in these unconquered regions is a
legitimate preliminary to exploration, 1 may say that I am
fully prepared, as Dr. Silberstein in one part of his paper
seems also prepared, to accept a gravitational influence on
the Ether's dielectric constant, and, therefore, on the square
of its index of refraction : though I should like to see this
done without postulating any increase of density in a
medium of which space is already completely full. It is
also highly desirable to avoid the frictional and thermal
considerations, accompanied by dissipation of energy, in-
separable from any sort of viscosity. These imperfections
are appropriate to a secondary or derived cosmic ingredient,
like matter; they are not appropriate to the fundamental
substance itself.
If the ether has demonstrated anything, so far, it has
shown us, by its veiy elusive character and complete
Structure for the Ether. 17&
transparency, that its properties are of the simplest and
most uniform kind, and free from any imperfections which
would accumulate waste energy in particular fractions of
itself. If we attribute all locomotion to matter in its most
general sense, including electric charges ; and all elasticity to
ether, regarding the latter as the uniting and potentially
strained medium responsible for every force which holds
atoms together ; we shall be on sound and simple lines. This
is not to deny that potential energy may be susceptible of
ultimate kinetic explanation, in terms of the postulated fine-
grained vorticity.
How the ether can be tied into the knots which we call
electrons â€” in other words, how the peculiar regions or
singular points characteristic of electric charge are consti-
tutedâ€” remains to be discovered. The small second-order
tension responsible for gravitation will, I feel surp, be
accounted for as soon as the electric structure is made out.
The fact that a luminous disturbance simulates the funda-
mental properties of matter is Giving us a broad hint.
A wave-front is an evanescent kind of matter â€” a sort ot
attempt of an accelerated electron to reproduce itself : the
question is how such a peculiarity, when generated, can be-
made permanent and its violent locomotion checked. We
must find out how to disturb the ether in such a way that
the modification shall remain concentrated, and not instantly
rush away and disperse itself wTith the speed of light. The
electric and the magnetic components must be separated,,
the one kept and the other annulled.
In this connexion, I take permission to make a few extracts from
the 1907 edition of ' Modern Views of Electricity,' so as to bring- the-
suggestions before those who may be interested in them. I quote from
pages 330 onwards : â€”
"Wherever electrons and atoms exist, they modify the ether in.
their immediate neighbourhood, so that waves passing through a
portion of space containing them are affected by their presence,
as if the ether were more or less loaded by them ; because the
electric displacements which go on in the unseparated and still
perfectly united constituents of free ether [in a beam of light] are
also shared to some extent by the separated peculiarities ....
A.11 those charges which possess externally-reaching lines of force
must share in the motion of the waves, without having the requisite
amount of resilience to compensate for their inertia."
â€¢'The positive and negative constituents, when they combine or
cohere, do not destroy each other and revert into plain ether again ;
on the contrary, they retain their individuality and persist, in either
a combined or separate state. We do not know how to produce-,
or to destroy these peculiarities .... [for whereas] matter can be,
174 On a Possible Structure for tlie Ether.
dissociated with extreme ease, the dissociation of ether is unknown
and hypothetical, save as represented by its apparent results.
"Nevertheless, it must be the case that the slight, almost infinites-
imal, shear, which goes on in a light wave, is of the nature of
incipient and temporary electrical separation .... It appears
possible that a sufficiently violent E.M.F., applied to the ether by
some method unknown to us at present, must be the kind of influence
necessary to shear it beyond the critical value and leave its
components permanently distinct ; such constituents being opposite
electric charges, which, when once thoroughly separated, only
combine to form matter, and do not recoil into ordinary ether
again."
Let me make one more quotation, immediately following, relating to
gravitation : â€”
"Every attempt at separation of this kind, even if no stronger
than exists in ordinary light', [is] accompanied by a longitudinal
force â€” Maxwell's pressure .... If the disturbance could be made
so extreme as to result in permanent dislocation, this pressure might
leave behind it, as permanent residue, a longitudinal pressure
[or tension] extending throughout space."
There seems to be a necessary connexion between transverse and
longitudinal stresses, the one being \ujv times the other. If we
re-estimate Maxwell's data for luminous vibrations, as given in his
article " Ether '' in the Encv. Brit. (Collected Works, vol. 2, p. 7(37), on
the basis of a reasoned high estimate of ether density, ignoring the guess
of that period that in the brightness near the Sun the amplitude of a
light vibration might possibty be as great as one-hundredth of a wave-
length, for this was only an upper limit and it is surely bound to be
much smaller, we can proceed thus : â€”
Let a be the maximum amplitude of shear of a light wave, y = a cos
p(x â€” vt), near the Sun; where the luminous energy is nearly 2 ergs
per c.c. ; and let u be the maximum speed of elastic recovery ;
then - =]hi â€” 2tt- .
v X
The energy \pu- = 2 ergs per c.c,
so, if p = 1012, u = 2 X 10~6 cm. per sec, and a = 10"17A.
Hence, expressing conditions near the sun in Maxwell's manner (/oc. cit.)7
Energy per cubic centimetre = ^py2a2// = 2 ergs.
Greatest tangential stress per sq. cm. â€” pv2ap â€” 6 X TO16 dynes.
Coefficient of rigidity of ether = pv2 = 1033 c.g.s.
Density of ether =p =1012 ,.
It will be observed that pv2ap is the same as pur, which is an
expression for the travelling momentum of a light-beam.
[ 175 ]
XVII. llie Spheroidal Electron.
Bi/ Prof. A. Anderson*.
ON the supposition that the shape of an electron in motion
is a spheroid, the direction of motion being along the
axis of symmetry, and the charge on the correlated electron
being distributed on its surface as if it were a conductor at
rest, the values of the momentum and energy in the gether
can be calculated. The length of the semi-axis in the direction
of motion is 1>, and that of the semi-axis at right angles to this
is a : b is thus the contracted length, or the length of the
semi-axis in the direction of motion after it has suffered the
Lorentz-FitzGerald contraction. As usual, /5 denotes the
quantitv (1â€” 2I , where v is the velocity of the electron
and c the velocity of light.
The results are, if fib>a.
momentum .
M=_ e*/3v r2/32b2-a2 1oo,/3/> + (/32A2-q*)-
1 6ttc2 (/32b2 -a2)hL /32b2 - a2 ' Â°g Â£/, _ (j&p _ tff
2 13b -j
(!32b2-a2)*J
and energy
If '&b
" ]â€¢
* Communicated by the Author.
m
176 The Spheroidal Electron.
If /3b = a, both expressions for M and both expressions-
for E lead to the same results :
This is the case of the Lorentz electron.
If we make a = b only the first pair of expressions can be
used as $b>a, and we obtain
M =
lbVac
fv2 + c2 , c4-v__2cl
L v2 * e â€” v v J '
C7ra \v c â€” v J
which are the momentum and energy associated with the
Abraham electron. Both electrons are, therefore, particular
cases of the general spheroidal electron.
The transverse mass is M/u, and well-known experiments
have been made to determine e/m and v, or, which is equi-
valent, e/M. and v. Thus it would, no doubt, be possible,,
though perhaps the mathematical work would be tedious,
to determine the value of the ratio of b to a for which the
theoretical value of e/M. would agree most closely with
the experimental results. A determination of this ratio
would bo of interest. We may, however, remark that
if the ratio of b to a tends to zero, the corresponding value
of M tends to
e2/3v
16ac2 ;
that is, the aether momentum associated with an electron
whose shape is a plane circular disk moving with uniform
velocity in a direction perpendicular to its plane is equal
to -^- times the momentum associated with the Lorentz
o
electron moving with an equal velocity. The ratio of
its transverse mass to its mass when v = 0 is the same as
for the Lorentz electron, and the experimental results
could not decide between them. In the case of a very
elongated prolate spheroid moving in the direction of its
axis of symmetry, both the momentum and energy become
very great.
[ 177 j
XVIII. The Adjustment of Observations. I.
By Norman Campbell, Sc.D*
1. X^OR more than fifty years the method of adjusting
J? observations affected by experimental errors has
always been that originally proposed by Gauss. The rules
necessary for its application are embodied in the formalism
of the c; Method of Least Squares/' Against the method
and the rules by which it is applied t\vo main objections
have often been urged : it is said that the theory on which
the rules are based is not true and that, even if it were
true, the rules are not an accurate expression of it. No
serious inquirer pretends nowadays that the method can
be completely defended against these objections : its use
is justified partly on the grounds of practical convenience
and partly on the ground that any method not open to these
objections would produce practically the same results. The
second contention is probably valid, but it provides a justifi-
cation for the use of the method only if the first is also valid,
and if there is no method equally convenient which is not
open to more serious theoretical objections. I believe that
the first contention is not valid and that in some cases â€” and
especially in those cases of most importance in physics â€”
there is a method of adjusting observations which is at once
more convenient in practice and more sound in theory. The
object of this paper is to explain and support that view.
Perhaps I may be pardoned for insisting at the outset
that my remarks deserve some attention. The theory of
errors has great intrinsic interest, but it is not a matter
to which physicists, even if they are in the habit of using it,
generally pay much attention. Its later developments are
extremely complex and highly technical, and most of us
do not study carefully the memoirs dealing with it which
appear from time to time in scientific journals. Those
memoirs do not generally pretend to subvert the accepted
rules for adjusting observations, but only to extend them
to somewhat unusual examples or to provide additional
support for them. But I do wish to subvert those rules :
I contend that the Method of Least Squares is an intolerably
cumbrous method for obtaining quite misleading results,
that there is a method which is incomparably simpler and
gives results which are not misleading, and that the only
persons who have any adequate reason for continuing the
* Commuicated by the Author.
Phil. Mag. S. 6. Vol. 39. No. 230. Feb. 1920. X
178 Dr. Norman Campbell on the
use oÂ£ the older method are the members of the National
Union of Computers (if there is such a body) who might
be thrown out of a job if the proposed method were adopted.
Since the only justification for the older method which has
so far stood the test of criticism is that it is practically
convenient, I maintain that the mere proposal of a more
convenient method throws the onus probandi on those who
refuse to use it.
2. The three problems.
There are three sets of circumstances in which the need
may arise for adjusting " inconsistent " observations.
(1) A number of measurements which do not agree
completely are made directly on a single magnitude, for
instance, the length of some definite rod or the time of
some definite process. It is required to determine from
them the " true value."" The rule universally adopted is
that the arithmetic mean of the measurements should be
selected as the true value*. We shall see that its validity
mio-ht be established directly by experiment. It is doubtful
whether the necessary experiments have actually been per-
formed, but I shall assume that the universal acceptance of
the rule shows that no experiments conflicting with it have
been made, and that, therefore, if it were suitably tested,
it would be established directly.
The matter is exceedingly important because on the
acceptance of this rule are based, either explicitly (as by
Gauss) or implicitly, the rules for solving the two remaining
problems. Any theory of error which is to lead to practical
rules must assume that in this case some rule is known for
determining the true value from the inconsistent obser-
vations. If we did not accept the arithmetic mean as the
true value, we should have to accept some other mean if any
progress was to be made.
(2) A number of measurements have been made on
several magnitudes between which a relation is known.
The arithmetic means of the measurements made on each
magnitude do not obey this relation ; consequently they
cannot be the true values. It is required to adjust the
observations so as to obtain true values which do obey
the relation. For example, the magnitudes may be the
three angles of a plane triangle : their sum must be 180Â° ;
* Some modification of this statement may be necessary if " systematic
error " is suspected. Such error will be discussed in the sequel. I do
not think it can arise if the conditions contemplated are fulfilled strictly,
and the measurements are made directly on a perfectly defined system.
Adjustment of Observations. 179
and yet it may be found that the arithmetic means of the
measurements made on each angle do not add up to 180Â°.
This problem is of great importance in some of the
practical applications of science, such as surveying It is
not of much importance in pure physics, for we very
seldom require to know with great accuracy and certainty
the value of any directly measured magnitude; it is
derived magnitudes that are important*.
(3) Measurements have been made on many sets of
magnitudes (x, y, z, . . .), which are known to be re-
lated by a numerical law of which the equation is
/(#, y, z, . . . a, b, c, . . .) =0, the form of the function f
being known, but not the values of the constants a, b, c, . . .
For example, measurements have been made of the activity
of: a pure radioactive substance (I) at various times (t). It
is known that I and t are related by the equation I = I0.e~M.
It is found that no values can be assigned to the constants
which are such that all the measured sets actually satisfy
the equation. It is required to determine those values of
the constants which are to be regarded as the true values.
This third problem is of immense physical importance,
and the solution or" it is involved in almost every expe-
rimental research. It is often solved by graphical methods :
numerical computation is used only when the number of
constants is too great to be represented on a plane diagram,
or when it appears that graphical methods do not utilize
fully the accuracy of the observations. But it is desirable
to discuss methods of computation applicable even to those
cases where graphical methods can also be used ; for it will
be admitted that both methods should be founded on the
same principles.
3. The principle of solution.
The accepted method oÂ£ solving the second and third
problems, which is embodied in the Method of Least Squares,
depend^ on the assumption that the true values of the mea-
sured magnitudes in the second problem or of the constants
in the thud are those which make the sum of the squares of
the residuals a minimum : the residuals are the differences
between the measured magnitudes and those calculated from
* The reason is that pure science is not concerned with the investigation
of the properties of individual objects, but only with the establishment of
laws. A magnitude which is determined by a law, and therefore important
for pure science, is always a derived magnitude.
N2
180 Dr. Norman Campbell on the
the true values. The rule for applying this method to the
third problem may be stated as follows : â€”
It is assumed that the equation which the measured
magnitudes have to satisfy is linear and of the form
aoc + ly + cz+ . . .+m = 0. .... (1)
If (as in the example of radioactive decay) the equation is
not of this form, certain methods (which will be accepted
for the present without inquiry) are available for reducing
it to this form. It is clear also that, without loss of
generality, we may always put a=l, and this procedure
will be adopted in what follows. If there are n variables
(#, y, z, . . .) and, consequently, n constants (b, c, . . . m),.
and if N sets of the variables have been measured, we have
N equations of the form
X\ + byl + cz1 + . . . 4- m = 0,"
x-s-h bt/js+ (:^-f . . . ' +m == 0.
(2)
N is greater than n. To obtain unique values of (b, c, . . . ?>/)
we have to reduce these N equations to n equations. We
form one of these equations by multiplying the rth equation
by xr and forming the sum of all the equations so multiplied ;
another by multiplying the ?*th equation by yr and forming
the sum ; another by multiplying the rth equation by zr ;
and so on. We thus obtain n "normal" equations relating
the n constants (b, c, . . . m) to sums of squares and products
of the variables (.i\ y,-zf. . .). In these equations (b, c, . . . in)
are now treated as variables; the solution of them gives the
true values of (6, c, . . . m) .
The rule for solving the second problem can be expressed
in a form very similar ; but since, as has been noted already,
this problem has not much physical importance, it will be
left on one side for the present. It will concern us only in
so far as we have to determine whether any other principles
proposed for solving the third problem are, like those of the
Method of Least Squares, also applicable to the second.
Regarded apart from the theory of errors on which it
professes to be founded, the rule given is merely a device
for reducing the K equations for n unknowns, the solution of
which must be indeterminate, to n equations for n unknowns,
the solution of which is determinate. But there is a much
simpler method of effecting the reduction. We may simply
divide the N equations into n groups, and add all the equations
in each group : we thus arrive at n " normal " equations.
Adjustment of Observations, 181
The procedure is so obvious that it would be the first to
occur to anyone to whom the problem was presented. It has
doubtless not been adopted mainly because the alternative
method of Least Squares was held to be the only one that is
justifiable by the theory of errors. But this procedure can
also be based on theory. When we select a group of the
equations and add them to obtain a normal equation, we are
assuming that this equation is absoiutel}' correct, and that
the sums of the measured magnitudes are the same as the
sums of the true magnitudes to which those measured
magnitudes approach. In other words we assume that, if
we take the sum of a group of measured magnitudes, the
â€¢errors of measurement cancel out; we assume that the sum
of the errors in any group is zero. Now if the group is
sufficiently large, this assumption will be true, even if we
believe in the Graussian law of errors, but it will also be
true if we adopt any other reasonable law of errors ; for
it is an assumption more fundamental than those on which
the Graussian law is based that positive and negative errors
are " equally probable." Accordingly, if the groups into
which the equations are divided are sufficiently large, the
assumption that their sum will be free from error is based
on theory much more firmly than the assumption of the
Gaussian law : for the first assumption is part of the second,
and the part which is least dubitable. The only question
which can arise is whether the assumption, and the pro-
cedure founded on it, is justifiable when the groups which
are added are not very large. In a later part of this paper
I shall argue that, even in this case, the procedure, though
not capable of complete theoretical justification, has more
theoretical justification than any other, and a great deal
more than that of the Method of Least Squares.
The proposed procedure may, therefore, be called the
Method of Zero Sum (Z.S.) in contradistinction to the
Method of Least Squares (L.S.). But even if its theoretical
basis is accepted, two further objections may be urged
against it. The first arises in connexion with the second
problem of the adjustment of observations. We have
measured the three angles of a triangle and find that the
measured values do not add up to 180Â°. The method of
adjustment proposed is to choose true values such that the
sum of the errors is zero. But it is at once apparent that
it is impossible to choose such values which will at the
same time add up to 180Â° : for the sum of the true values
must be the same as the sum of the measured magnitudes :
in this example then the method will not work. 1 cannot
182 Dr. Norman Campbell on the
help thinking that it is the failure of the method in this
important application which has prevented its serious con-
sideration. I shall argue later that in this example the
Gaussian method also will not work, and that its application
in the ordinary manner leads to a result which is directly
contradictory to the assumptions involved in it. The
Gaussian method, like that of Z.S., involves the assumption
that there is no systematic error, and in this case the
assumption cannot be maintained. We must base our
procedure on a theory which recognizes systematic error.
However, at present we are concerned only with the third
problem, and in connexion with this an objection may be
urged against the method of Z.S. It may be said â€” and of
course the statement of fact cannot be disputed â€” that the
result which will be obtained will depend on the manner in
which the observational equations are grouped to obtain
the normal equations : if one grouping is adopted, one
set of values will be obtained ; if another grouping is
adopted, another : Z.S. does not, like L.S., lead to an
unique solution. Herein lies, to my mind, one of the
chief advantages of Z.S. For the apparent uniqueness
of the solution by L.S. is altogether misleading. It is
true that the introduction of " probable error " admits
implicitly that solutions other than that given by the
normal equations are admissible ; but that admission is
so important that it ought to be stated explicitly. When
we discuss in detail the theory of the matter we shall see
that there is not the slightest reason of any kind for
selecting one of the admissible solutions rather than
another. However, for the purposes of practical con-
venience, it is certainly desirable to have some standard
method of selecting a single value to represent the obser-
vations, even if that which is selected is not really different
in importance from many others, simply in order that no
scope may be left for personal choice and that all persons
who consider the same observations may arrive at the same
single value for expressing them. But if it is admitted
that the choice of that standard method is to a large extent
arbitrary, there is no difficulty in devising one that is
suitable.
Accordingly in the* application of the method of Z.S.
it is proposed that the groups should be selected in the
following manner. There is always at least one of tho
measured magnitudes (x, y, z, . . .) which may be assumed
to be free from error : let it be y. (This assumption is also
involved in L.S.) The observational equations are to be
Adjustment of Observations. 183
arranged in increasing (or decreasing) order of y. If there
are N equations and n magnitudes (x, y, z, ... .), and if
N=/m-f q, where p, q are integers, then the first normal
equation is to be formed by adding the first p equations in
this order, the second by adding the second p equations,
and so on until nâ€”q normal equations have been formed;
the other q normal equations are to be formed by adding p + 1
observational equations taken in order in the same way *.
The basis of part of this rule is obvious. The method
provides that the number of observational equations added
to form a normal equation is (as nearly as possible) the
same for each normal equation. Since the assumption
underlying the method is only true if that number be large,
it is desirable to prevent it being smaller than it need be in
any one case ; that result is obtained by making the number
equal in each case. The basis of the remainder may be
seen by considering the case when there are only two
magnitudes, x, y. Then we are practically taking the
mean of each of two halves of the observations and deter-
mining b and m from these two means. The determination
might be made graphically : we might plot the points
representing the two means and draw a straight line
through them. It wTould then be obvious that the accuracy
with which the straight line could be drawn would be
greater the greater the distance between the two points.
It is desirable therefore that the difference between the
two means should be as great as possible : this condition
is obtained by arranging the observations in the order
proposed for the purpose of forming the normal equations ;
for one mean is that of all the smallest values of y and
of all the smallest (or largest) values of x, whereas the
other is that of all the largest values of y and all the largest
(or smallest) values of x.
4. Errors.
It will be convenient also to express the matter ana-
lytically. The normal equations will be
X1 = bYl + cZ1+ ..
X2 = 6Y2 + cZ2+ . .
m> ' (3)
J
* It may be observed that if q is not 0 the result obtained will be
slightly different according as an increasing or decreasing order of y is
adopted. But the differences arising from this latitude of choice are
completely negligible.
184
Dr. Norman Campbell on the
where (X, Y, Z, . . .) are the means * of a group of observed
The solution is
b = DÂ»/D ; C = Dc/D ; . . . ; m = Dw/D ; . (4)
rhere
D* =
XxZj.
. . 1
; D.=
Yx Zi â€¢ â€¢
.*1
: D =
YxZx.
. . 1
â– J\:2 ^2 â€¢
. . 1
Y2 Z2 .
x2
Y2 Z2 .
. 1
â€¢ â€¢
â€¢
â€¢ â€¢
â€¢
â€¢ â€¢
â€¢
Now suppose that only one of the measured magnitudes, #,
is liable to error amd that all the measurements on the others
are absolutely correct. This assumption is practically true in
a large number of important cases ; it is moreover essentially
involved in the method of L.S., so that we are not introducing
any new error by adopting it. Then if db, de, . . ". dm are the
errors caused in the calculated values of "6, c, . . . m, by errors
dXl5 & = 9 ; ffm= -y 2_y 2' ' * ^ '
If there are three magnitudes and three constants b, c, m
we have
(Y,Z, -Z1Y2 + Y8Z3-Z2Y,+ Y3Z1-Z,Y1)S
^- (Z1-ZI)Â« + (ZÂ»-Z# + (Z1-iyÂ»
_ (Y^â€” 2^2 -fl^Za â€” Z2Y34- Y3Z1 â€” ZsIlj)2 ._>
P'~ (Y.-YO'+tY.-Y^ + C^-Y,)' ;
(YiZi-Z^ + Y^-ZsYa+YsZj-Z^!)2
'" ~ (-YA-ZxY,)' + (Y2Z3 - Z2YS) 2 + ( Y^Z, - ZSY,)2 J
From (6) we reach again the conclusion that the calculated
values will be made most accurate by making Y1 and Y2 as
different as possible, which is effected by grouping the obser-
vations in the manner proposed. In (7) the matter is more
complicated, and the most accurate way of grouping the
observations depends on the values of b, c; but in many
important cases the rule which has been proposed gives
the calculated values the greatest possible weight, and in
no case does it seem to give them a weight very much less
than the greatest possible. When ease of application is
taken into account it is improbable that any more suitable
rule of general validity could be found.
By the aid of (4) and (5) probable errors of the calculated
values can be found in a manner exactly similar to that of
the method of L.S. We shall inquire later what is the
significance of such probable errors according to L.S. or Z.S.,
but for the present we shall use them merely as a rough
method of comparing the results obtained by the two methods.
It should be observed that when (4) is used in L.S. to obtain
the probable error, dx2 occurs in place of dX2, where dx2 is
the mean square error of a single observation and dX2 the
mean square error of the arithmetic mean of p such single
observations. In our estimates of probable error by Z.S.,
186 Dr. Norman Campbell on lite
we shall assume the usual results of L.S., namely, that
2y
dx2=â€” , where Xv2 is the sum of the squares of the
JN â€” n x
dx*
residuals, and dX2 = â€” T.
p-1
Examples.
It will now be well to show how the proposed method
works out in practice. The chief difficulty is in the
selection of material. I have a large amount of matter
of my own to which it might be applied, but the use of
that matter might not inspire confidence. Observations
are not usually published in sufficient detail to enable them
to be recalculated ; so three examples have been taken
(although they are not wholly suitable, because the number
of observations are so small) from the 8th edition of
Merriman's ' Method of Least Squares/ pp. 126, 132, 138.
The advantages of the method in the saving of labour increase
rapidly both with N and n, but even in these simple cases
they are enormous. In L.S. Â£Nn(n + 1) multiplications have
to be performed, and then ^(Â« + 2)(n + l) columns, each of
N entries, added ; in Z.S. there is no multiplication, and
only ~N(n + l)/n columns, each of N/w entries, are added.
Using a calculating machine, to which I am thoroughly
accustomed, omitting all " checks " (and the omission wastes
time on the whole), and reducing writing to a minimum by
keeping figures on the board of the machine, I found that
the mere writing, quite apart from calculation, involved in
the formation of the normal equations of L.S., occupied
longer than the complete formation of these equations
by Z.S. The solution of the equations takes the same time
in either case : it takes longer than the formation of the
normal equations by Z.S., but not nearly as long as that
by L.S.
In each example there is given (1) the equation which the
observations have to satisfy ; (2) the observations ; (3) the
normal equations and solution by L.S. ; (4) the normal
equations and solution by Z.S. ; (5) in the last four columns
of the observations, the residuals and their squares according
to L.S. and Z.S. The observations which are added to give
the normal equations of Z.S. are bracketed. In the third
example the equation (1) is not linear : for the purposes of
calculation it was reduced to linear form, in accordance with
the usual practice, by taking logarithms of both sides ; these
logarithms, used in the calculation, are given in the table.
Adjustment of Observations.
187
Example 1.
x = by + m.
L.S.
A
Z.S
, *
y-
r^
""">
r
v X 105.
^2xio8.
UX105.
y'XlO8
(1)1
3921469
0-9688402
+ 318
1011
+324
1050
(2)
39-20335
0-9289304
- 9
1
- 4
0
.(3)
39-19519
0-8904120
- 49
24
- 42
18
(4)
> 39-17456
0-7929544
-281
790
-277
767
(5)
39-13929
0-6127966
- 26
7
- 23
5
(6)
39-10168
0-4254385
+ 1
0
+ 1
0
(7) j
3903510
0-0948286
+ 25
6
+ 23
5
(8)"
39-02425
0-0505201
-164
269
-167
279
(9)
39-01884
00341473
-374
1399
-377
1421
(10)
39-01997
0-0218023
- 12
1
- 14
2 .
(11)
* 3902410
0-0190338
+457
2088
+454
2oeT
(12)
39-01214
0-0019464
-393
1544
-396
1568
(13)j
39-02074
00000515
+ 505
2550
+ 503
2530
2v+ 1306 2
;2 9689
2v+ 1305 2i
2 9706
2v_ 1308
2t>_ 1300
(L.S.) 508-18390 = 13-000000 Â»i + 4-848702 6
189-94441 = 4-848702m + 3-804394 6*
m = 39-01568 Â± 0*00077, b = 0-20213 Â± 0-00142.
(Z.S.) 274-06386= 1m
234-12004= 6m
39-01571 + 0-00089, b
m
+ 47212016
+ 0-1275016
0-20204 + 0-00184.
From eqns. (1-7) m = 39*01530 b = 0*20265
â€ž (8-13) m = 38-9751 b = 0-21118
t This equation is misprinted in Merriman.
In Examples 1 and 3 there is no material difference
between the results obtained by the two methods ; they
agree well within the probable error. The probable error
in Example 3 is actually less for the result by Z.S. than
for that by L.S. Indeed in that example the comparison
does not appear as favourable to Z.S as it ought to be. For
the residuals are calculated for log x : strictly they should
be calculated for x. If they are so calculated, %v2 is slightly
less for the result by Z.S. than for that by L.S. : the Least
Square method does not actually produce the least squares.
188
Dr
Norman Campbell on the
E
XAMPLE 2.
x â€”
by-\-cz-\-m.
L.S.
Z.S.
X.
y-
z.
r~ >*
r~ A
vxio4. ^xios.
uXlO4. v2Xl08
(1)1
(2) \
8-1950
0
0
- 1 1
+ 25
G25
3-2299
01
001
--18 324
-14
196
(3) ]
3-2532
0-2
0-04
4-2 4
-11
121
(4) |
3-2611
0-3
009
+21 441
- J
1
(5)
32516
0-4
0-16
+ 19 361
- 6
36
(6) 1
3-2282
0-5
025
+ 31 961
+ 8
64
(7)]
(8) |
3-1807
0-6
0-36
-44 1936
-59
3481
3-1266
0-7
0-49
-33 1089
-33
1089
(9) j"
(10) J
3-0594
0-8
064
0 0
+22
484
2-9759
0-9
0-81
+ 24 576
+72
5184
2iÂ»+ 97 2v25693
2u+127 2v2
11261
â€¢Ev- 96
2v- 129
(L.S.) 31*761600 = 10-00m + 4-500# + 2-8500*
14-089570 = 4-50m + 2-850?/ 4 2*02502-
8-828813= 2-85m + 2-025y+ 1*5333*
m = 3-1 951 + 0-001-5, b = 0-4425 + 0-0077, c = -0-7653 Â±0-0081.
(Z.S.) 9-6781 = 3m +0% +0-05.;
9*7409 = 3m +l'2y + 0*50*
12-3426 = 4m + 3% + 2*30r
m = 3-1925 + 0-0034, Â£ = 0*4678 + 0*0072, c = -0*7960 + 0-0081.
From eqns. (1-5) t m = 3*21955, b â€” 0*3652, c = -0'7079.
(6-10) m = 3-20265, b = 0*2670, c- -0*3065.
This discrepancy is, oÂ£ course, due to the fact that it is the
sums of the squares of the residuals of loga?, and not of x,
that have been made a minimum : this process, though
almost always adopted, is not justifiable by the theory on
which the method is based. On the other hand, it is
legitimate to apply the method of Z.S. to the logarithms ;
for, so long as the errors are small, the solution which
makes the sum of the errors of x zero will also make
the sum of the errors of f(x) zero, whatever may be the
function / (so long as it has no singular points in the
neighbourhood). Here is another advantage of the method
of Z.S. It can be applied directly to an equation ihat
is not linear, so long as that equation can be reduced to
A djusi
ment of Observations
189
Example
3.
x = myi
logy.
L.S.
Z.S.
U-
log X.
yxlO4. v-
XlO8.
VXlOl
V
jxio8
(1)1
1-731
0-1144
0-23830
-0-94157
-f 7
49
+ 9
81
(2)
1-853
01312
0-26788
-0-88207
-38
1441
-37
1369
(3)
> 1-984
01445
0-29754
-0-84013
+ 18
324
4-19
361
(4)
2081
0-1579
0-31827
-0-80162
4- 5
25
+ 5
25
Â®L
2-171
0-1701
0-33666
-076930
+ 4
16
4- 3
9
(7)
2-258
0-1813
035372
-0-74160
4-15
225
+15
225
2-326
0-1925
036661
-0-71557
â€” 5
25
- 6
36
(8)
2-397
2-460
0-2026
0-37967
-0-69336
- 1
1
â€” 2
4
(9).
0-2123
039094
-0-67305
- 5
25
- 7
49
2v+ 49 2v2
2134
2y+ 51
2v
'2159
2v- 49
2w_ 52
(L.S.)
(Z.S.)
2-9496 = 9-0000 loo- m
0583Z/
-2-2758 =-7-0583 log Â»i + 5-6008 ft
m = 5-98 Â±0-050, b = 0-5727+0-0045.
1-45865 = 5 loo- m -4*23471 b
1-49094 = 4 log m- 2-82358 b
m = 6-004-0-025, b = 0'5743Â±0-0023.
From eqns. (1-5)
(6-9)
m = 6-07,
m = 5'7o,
b = 0-580.
b = 0-546.
the linear form by a change oÂ£ variable ; whereas strictly
the method oÂ£ L.S. ought not to be so applied. If the
method of L.S. is adopted the process of successive approxi-
mation, which has to be used when the equation cannot be
reduced to the linear form by change of variable, ought
always to be applied to a non-linear equation, even when it
can be so reduced.
On the other hand, in Example 2 there is a difference
between the values of b and c deduced by the two methods
which exceeds the probable error of either method. But an
examination of the residuals at once shows the explanation.
The equation to which the observations are to be fitted
is empirical; the residuals show a systematic variation
which indicates that the formula is not strictly true. At
the bottom of the table are given the values of b, c, m
calculated by L.S. first from the first 5 equations and
190 Dr. Norman Campbell on the
second from the last 5. The differences between the
resulting values of b and c obtained from these two parts
of the observations differ by far more than the probable
error ; this difference is an indication that the formula
applicable to the observations with small values of y is
not applicable to the observations with large values. It
is not strictly legitimate to apply the same values of b, c, m
to all the observations. And if: it is not legitimate, of
course the method of Z.S., which divides the observational
material into two parts and assumes that the equation is
equally true for all of them, will give a result different
from that obtained by the methud of L.S. which treats
that material as a whole. If our object is merely to
represent the observations as nearly as possible by a con-
venient empirical formula, it may certainly be better in
such cases to employ the method of L.S. : a closer fit is
likely to be obtained. But then the problem is not one
of pure physics, which is not concerned at all with merely
empirical formulae ; neither method of adjustment has in
such cases any true validity at all, for the adjustment itself
is fundamentally false. And if the object is merely to
obtain an empirical formula it may be urged that, though
L.S. gives a closer fit, it is scarcely worth obtaining at the
expense of the enormously greater labour.
It may be noted that, for comparison, the observations of
Examples 1 and 3 have been divided similarly into two parts,
each of which is adjusted separately by L.S. The results
are given at the foot of each table. It will be seen that in
these examples, where the equations do fit the observations
(although the equation of Example 3 is also empirical), there
is very much less difference between the values given by the
two parts of the observations. (The value of m for the second
half of Example 1 is subject to a very large probable error,
as may be seen by examining the observations on which it is
based ; the differences are not at all inconsistent with the
applicability of the same constants to the whole of the obser-
vational material.)
5. Some further considerations.
These examples â€” and a great many others have been
-examined â€” show that the method proposed is perfectly
practicable and that it does not lead to results differing
very greatly from those of the method of L.S. And that
proof, as I hold, throws the task of justifying their action on
those who continue to emplo}^ a method which is admittedly
invalid theoretically and exceedingly cumbrous in practice.
Adjustment of Observations. 191
I maintain that, until it is shown to lead to misleading
results, the method of Z.S. holds the field against any other,
merely on the ground of simplicity. However, I am pre-
pared to admit that in certain cases it loses much or all oÂ£
its advantage, namely in those for which N is not very much
greater than ?i, and the number of observations not much
greater than the number of variables. It is then found â€” as
might be expected from the fact that p is little if at all greater
than 1â€” that values obtained for the constants vary greatly
with the precise grouping of the observations in the formation
of the normal equations, and the probable error of the result
is much greater than can be judged significant according to
the criterion which will be developed at, a later stage. Of
course the best method of dealing with such cases is to make
more observations and so cause N to be much greater than n ;
but if, for any reason, that course is impossible, and if some
single value must be obtained, then it is probably better to
employ L.S. unless N is at least as great as 3Â». But the use
of that method is a mere matter of practical convenience :
I deny altogether that, in general, the results obtained have
any greater theoretical significance than the widely differing
results obtained by the method of Z.S. There is simply no
theoretical ground for any single value whatever within very
wide limits.
These considerations have a bearing on the second problem
of adjustment of observations, namely that in which it is
required to determine true values of the measured magnitudes
and not constants of an equation which they satisfy. The
true values are now values such that they satisfy some
equation of which the constants are definitely known. In
one form of the problem, this equation contains, besides the
variable magnitudes, a constant term to which a definite
numerical value is assigned. An example of this form is
the problem of the angles of a triangle, and we have already
noted that the method of Z.S. (and that of L.S.) must fail
when applied to that problem. But in a second form, the
" equation of condition " does not contain a constant term,
but relates only the measured variables. An example of
this form is the problem of a< I justing the results of a
levelling survey : here it is known (e. g.) that the height
of A above C must be the sum of the heights of A above B
and of B above C, but there is no numerical constant known
apart from the observations.
In this problem it is possible to find true values such that
thev satisfy the equation of condition and make the sum of
the errors zero; and rules for applying the method of Z.S.
192 Dr. Norman Campbell on the
can easily be devised. But since the equations of conditions
are never much more numerous than the observations (they
are usually much less numerous), N is never large compared
with n. Accordingly Z.S. gives results varying widely as
different methods of grouping are adopted ; and though
there is no reason to believe that all these results are not
admissible, ifc is difficult to fix precisely one method of
grouping, so that the first requisite in problems of surveying,
namely that a definitely unique set of values shall be obtained,
is fulfilled. It is certainly better to adjust by the method
of L.S. And it is fortunate here that the method loses most
of its disadvantages. The coefficients of the equations of
condition are usually small integers, so that the calculation
is easy. Moreover it. will appear in the sequel that this is
the one form of problem to which L.S. is strictly applicable
on theoretical grounds. It is here that there is most evidence
that the Gaussian law of error is true, and here that the
method is an adequate expression of the Gaussian theory.
I believe indeed that the Gaussian method was originally
elaborated to deal with just this problem : if so, it was
completely justified. It is only its extension to the other
form of the second problem (where there is a constant
term in the equation of condition), and to the third
problem, that is both theoretically illegitimate and prac-
tically inconvenient.
It is admitted then that, in this direction, room still remains
for the method of L.S. This admission may seem to weaken
somewhat the case for its replacement elsewhere by Z.S.
Accordingly, before proceeding (in a subsequent paper) to a
discussion of the validity of the two methods according to
the theory of errors, it may be well to point out that there
are examples to which L.S. is as clearly inapplicable as Z.S.
is to that which has just been discussed. These examples
occur when the equation (1) reduces to the simple form x = by,
as happens when we have to determine a density (b) from
measurements of a mass (x) and a volume (y). .
An elementary student, when he had measured several sets
of associated values of x and y, would doubtless take the ratio
cejy in each set, and take for b the mean of these ratios. A
more experienced worker wrould realize at once that such a
procedure gives undue w7eight to the raiios derived from the
smaller values of x,y, which are likely to be less (relatively)
accurate. He would probably add all the #'s and all the ?/'s
and take the ratio of the sums. This is exactly the procedure
of the method of Z.S. But nobody, I believe, would adopt the
J
Adjustment of Observations. 193
method of L.S., according to which 6=^â€”. In this case
Â° z the angle between the d'-axis and the
tangent at D to the central line of the beam, so that d + dT = 0, (71a)
whence
dT = _Gd$
dx dx
But dy -' â– , ,
â– â€” = tan (p = 9 nearly.
clx
Therefore
Â£â€” g3.- ..... (71)
dx dx1
The moment about the tangent at J)' of the shearing force
at D, as well as of whatever load there is on the element
DD', nas been neglected in the preceding equations because,
assuming that the load is on the central line of the beam,
this moment is a quantity of at least the second order. The
correct equation when the load is not on the central line is
given later (equation 101).
Now the bending moment G can be resolved again into a
pair of components about lines respectively parallel and
Buckling of Deep Becu
197
perpendicular to the twisted depth at D (see fig. 10). The
former component, of magnitude Gtt, bends the central line
in a plane perpendicular to the depth of the beam, and this
plane of bending is everywhere nearly horizontal. If there
Fig. 10.
is an end couple M on the beam acting in a horizontal plane,
as in clamping the end, then the total couple at D causing
bending in a horizontal plane is Gt+M.
Then, since the curvature produced by this couple is the
cause of the deflexion y, we get
EC^ = Gt+M.
doc2
(72)
Equations (71) and (72) are the general differential
equations, which, together with the end-conditions of the
beam, determine the buckling load when there is no tension
or thrust in the beam, and when the load is applied at the
centre line of the beam. â„¢
Substituting in (71) the value of -y-^ from (72) wre get
dT
dx
^(Gr + M).
(73)
From the meaning of Kn we have
KnX (angle of twist per unit length) = torque,
that is,
Knâ€” =1.
ax
Therefore (73) becomes finally
G(Gt + M).
(74)
(75)
It is clear that a tension in the beam would help to
stabilize it, and that a thrust would make it less stable, for
the beam could buckle under a thrust alone. If a thrust R
is applied at the ends of the beam, then an extra term â€” R#
198 Dr. J. Prescott on the
occurs on the right of (72). Thus
EcJ=Gt+M-%,
(76)
the term liy being the bending moment in Euler's theory of
struts.
Case 8. â€” Beam of length I under a pair of balancing
couples, each G, at the ends, together with a thrust II
(fig. 11).
Fig. 11.
B
Eievatson
B
D
Plan of Central Line
It is understood that the section of the beam has at least
one symmetrical axis which is assumed to be vertical, and
the length of this axis is several times as long as the greatest
horizontal breadth of the section. The end couples men-
tioned in this problem are in a vertical plane parallel to the
length of the beam.
It should be noticed that the direction of G, and therefore
of t, are contrary to their directions in Case 1 in the first
paper.
In this case M is zero. Then the equations applying to
this case are (71) and (76).
Since G is constant (71) gives
(77)
or
Tâ€” Gg + N,
ax ax
(78)
But at the middle of the beam r and y have both maximum
or
minimum values, and consequently -*- and -~ are both
.zero. It follows that the constant N is zero,
Buckling of Deep Beams. 199
Integrating again we get
Knr=-G//, (79)
no constant being added in this case because both r and y
are zero at the ends. The negative sign on the right side is
due to the fact that t is negative by our convention.
Now equation (76) gives, since M is zero,
eoS=-{Â£+r>' â€¢ â€¢ â€¢ <*Â»
which is the same equation as for a strut under a thrust
/G2 \
I p â€” \- R I . By exactly the same reasoning as in Euler's
theory of struts instability occurs when
Â£+R)=ECS
From this it follows that a very big couple could be
applied at each end and the beam would remain stable
provided a suitable tension R' is also applied. That is, the
tension wholly or partially neutralizes the effect of the end
couples in producing instability, while the couple G weakens
the beam when used as a strut.
Case 9. â€” Beam under the same forces as in the last case
with the addition of a pair of couples in horizontal planes
applied at the ends to keep -j- zero at those points.
This corresponds to a strut with clamped ends, and differs
200
Dr. J. Prescott on the
from Case 2 only in having a thrust in addition to the
couples.
Equation (79) is true in this case as in the last. In
equation (76) M is not zero for this case, and therefore the
equation corresponding to (80) is
ec3=-{Â£+r>+m â€¢ â€¢ â€¢ Â«*>
This is the same equation as for a strut with clamped ends
(P2 \
jr â€” KR I . Also the rest of the conditions for
the beam are the same as for the strut. Therefore, by
analogy with the strut,
G2 , ^ 4tt2E(J
tt +K= â€” y â€”
i\ n r
(84)
If a tension R' is applied instead of the thrust R the
equation becomes
4tt2EC
It should be observed that the two cases just worked out
can be regarded as solutions of the strut problem. Suppose
the two thrusts at the end are each applied at distance p
from the centre of the end section in the direction parallel
to the depth as in fig. 12. Then the couple G is Rp and the
Fig. 12.
~ T
P
thrust at which instability begins for a pair of free ends is R
given by equation (81), that is, by the equation
ivn
EC
7T
(85)
If Rp2 is small compared with Kn we may use the approximate
equation
R = EC^-Â£(EC5)2
= EC
7T2(, TrVEC
m1-
VEC],
/2 Kn J *
Buckling oj Deep Beams.
201
Without assuming that R is small we see that equation (85;
gives a pair of roots with opposite signs. The negative root
indicates a tension, and thus we see that the beam could
buckle under a tension applied in a line parallel to the
unstrained central line but not along it. Since the sum of
the roots of the equation is negative, it follows that the
tension that would cause buckling is greater than the thrust
that must be applied in the same line. Moreover, if p is^
small then the tension at which buckling occurs is very
great, and an approximate value is obtained by dropping the
term on the right of (So). This approximate value is
R=
is
This is a very remarkable result in that this tension
independent of the flexural rigidity of the beam.
the first paper the loads were all taken on the centre
In
of the sections of the beam. Two cases will now be worked
out to show the effect of taking a load slightly off the centre
of the beam.
Case 10. â€” Beam built into a support at one end and free
at the other where a load P is applied.
Plan of Central
The load P, before the beam is strained, is situated at
(o, p, q) referred to three rectangular axes through the free
end, as indicated in fig. 13.
202 Dr. J. Prescott on the
The origin being at the free end of the beam the bending
moment at x is
G = P#,
and therefore equation (75) gives
rf2T
ECKn~=-PV2T,
dx
or d2r
da?
where Â± P
= â€” m4x2r, . . . . (86)
*
EraCK
(87)
This is exactly the same equation as for Case 3 in the first
paper. The only difference is in the end conditions. These
conditions are now
t=0 where x â€” l, (88)
â€” Kn-r- = torque
= F(9T+P) where a? = 0. : . (89)
The negative sign is necessary because r decreases as x
increases.
The solution of equation (86) in series is
f m*xh m8x9 ~]
T=gt/-0- + 4.5.8.9-J
+ Hl-HT4+ 3TOT8 i">- â€¢ â€¢ (90)
Now condition (89) gives
Kna=-F(qb+p), ..... (91)
and condition (88) gives
+Â»{'-S+n?n!--};- â€¢ â€¢ Â«
Substituting the value of a from (9.1) in equation (92),
and writing 5 for mH*, we get
Fljqb+p) t t I
Km X 4.5^4.5.8.9 â€¢â€¢"(
-6{1-374 + 3^7T8- }=0-<93)
Buckling of Deep Beams. 203
Now the value of P which makes b infinite will make t
infinite except at the fixed end. That is, the beam is
unstable for that value of P which makes the coefficient of b
zero in equation (93). The condition for instability is,
therefore,
.gfr/iâ€” 1_ + *2 i
KÂ«V 4.5^4.5.8.9 /
+ {1-ri-+Tror8-l"0'-' (94)
It should be observed that this equation is independent of
p, which shows that the stability is not affected by putting
the load a litttle to one side of the centre. The only result
of displacing the load laterally is to put a torsion on the
beam, thus giving a new equilibrium state of the beam from
which instability begins.
Let us writeâ€ž for shortness,
/w=i-A-Â»
4.5 4.5.8.9
FÂ« = 1-^t+,
3.4 ' 3.4.7.8
Then our equation for the load is
-g/M + F(*)=0 (95)
An approximate solution of this is the solution of the
equation
F(Â«)=0, (96)
since the other term is small because q is small.
Let 8X denote the solution of (96), and let Px be the
corresponding value of P. P2 is, of course, the value of P
found in Case 3. In the small term in equation (95) we
may use the approximate value Px for P. Then writing 8
for x?-^> and (s^-j-z) for s, equation (95) becomes
-Sfl*i + Â«) + Â£(* + *)=*<>â€¢
Since z is small,
F (s, + z) = F(sj ) -f zF'(si) approximately
Then taking account of the first powers of 8 and z only
we get
-S/W+*FlÂ«i)Â«0.
204 Dr. J. Prescott 'on the
Now
â€” .i-/i_ * , i
3.U 4.7-4.7.8:11 J*
and, from the result of Case 3,
Si=eS=4'0122=16'096-
The results of tedious arithmetic are
7(50 = 0-3577,
Â¥/(s1)= -0-04344.
Therefore
0-3577
'~ 0-04544 b
â€” *23PA
Kn
Hence
and
= 5, 4-S.
\Zi=1+i71
2 y/Sl \'Sl
Finally
1 / 8-23P1/gv'EnOK\
" 2 x 4*012 \ " P^Kn /
PZ2= ^X VEnCK
= 4-012 [ VE^CK- 1-025 ISC \ . . (97)
The critical Euler load for this beam, when used as a strut
and supported in the same way (fig. 14), is
B=?5 - W
Buckling of Deep Beams.
205
Now it is worth while to note that the correction to the
buckling load due to putting the load at height q above
the centre of the end instead of at that centre is
4-012x1-025
q EC _ 4-012 x 1-025 x 4 y-p
I I2 ~ 7T2 7
= 1-666 1 R,
(99)
thus showing once again the intimate connexion between
the strut problem and the buckling beam problem.
Fig. 14.
Mr
Case 11. â€” Beam under a total load W distributed as a
uniform load w per unit length, the load at x being situated,
before the beam is strained, at (#, p, q). The beam is fixed
iit one end and free at the other as in the last case.
We have now to extend equation (71a) to the case where
the distributed load is not on the central line.
Dealing with the element in fig. 9 we find that the load
wdx, when the beam is twisted, has a torque wdx(qr+p)
about the tangent at D'. Then, instead of (71 a), we get
GW^ + rfT.+ iwte(5T+p)=0, . . . (100)
and instead of (71)
dT {i dh.
which, when M = 03 as in the present case, is equivalent to
KÂ»S=-&-(^)- â€¢ â€¢ â€¢ (]o2)
206 Dr. J. Prescott on the
This is the general equation when the load is off the
centre and there is no couple M at the ends. In the present
problem, the origin being taken at the free end, G- = ^wx2,
and therefore
Whether p and q are functions of x or constants the
solution has the form
T = a0f(x)+alT(x)+cf>(x), . . . (104)
where (l).
Eliminating a^ from these we get
0 = a0{/'(0)F(Z)-/(/jF'(0)}
+*(0)F(i)-*(J)F'(0). .... (105)
Now the analytical condition for instability is that a0
should be infinite, and this can only occur if the coefficient
of a0 in (105) is zero ; that is, the condition for instability is
/'(0)F(Z)-/WF'(0) = 0, .... (100)
which is independent of and therefore of p.
We have now shown, as in the last case, that p has
nothing to do with stability.
Since p does not affect stability we can drop it from our
equations. Then, assuming that q is constant, equation (103)
becomes, when p is dropped,
^ = -(mÂ¥ + ^)T, .... (107)
where 6 w1 maw,
m~mm> (10b;
r,_ wq
f~ Kn
(109
Now the assumption is being made that q is small, and
Buckling of Deep Beams. 207
the problem is to find t when q is zero and then correct
Â£01* q (or c2).
When c is zero the solution of (107) that satisfies all the
conditions of the problem is
^41-5T6- + 5T6TTT7I2-}' * (110)
m being given by the following equation, taken from Case 6
in the' first paper,
m6Z6=41-30 (Ill)
The value of t in (110) being denoted by rx the equation
we have to solve is approximately
^+wVt=~cj2t1, .... (112)
the term on the right being now regarded as a known
function of a. This process amounts to treating c4 as
negligible while c2 is not negligible.
The particular integral of (112) is
2 , f 1 mV/ 1 1 \
Ml12*18 /' . 1 1 1 \
+ 13. UU.2.7.8 + 5. 6. 7. 8 + 5. 6. 11. 12/
_ m1 V8 / 1 1
19 ". 20 \1 â€¢ 2 . 7 . 8 . 13 . 14 + 5 . 6 . 7 . 8 . 13 . 14
+ 5 . 6 . 11 . 12 . 13 . 14 + 5. 6. 11. 12. 17. IS/*" ')
= -a0A2F(mV5),say (113)
Then the complete value of r that satisfies the condition
that the torque is zero at the free end where x = 0 is
T = T1-a0cVF(mV)
= a0/(mV)-a06Â¥F(m6^, . . . (114)
where /(m6^6) is the series in the brackets in equation (110).
The other condition that has to be satisfied is that r=0
when x = L Therefore
0 = i\mnG)-c2l2F(ni6l6) (115)
208 Dr. J. Prescott on the
Let s be written for the value of w6/6 satisfying
/(m6Z6)=0, ...... (116)
that is, 5 = 41-30, (117)
as given in equation (111). Then let the solution of
equation (115) for >?i6/6 be written
mGl6 = s + v.
Actually equation (115) has to be solved for w which is
involved in both c and m, but since the term involving c is
small, we can use, in the expression for c, the approximate
value of iv given by (111).
Now equation (115) becomes
0=f(s + v)-ciPF(s + v)
=f(s) + vf(s)-c*FF{s)
â– as far as terms of the first dimension in v and c2.
But/(Â» = 0by (116). Therefore
Since
(118)
^S) ~ "~ 5~6 + 5 . 6 . 11 . 12 ~ 5.6.11.12.17.18 + '
therefore
â€žn_ 1 2s 3s2
* [S)~ 5.6 + 5.6.11.12 5. 6. 11. 12. 17. 18 +'
With the value of s given in (117) we find that
and F(s) = 0-1888.
Therefore _ _ 0-1888 x 30
C~ 0-4890 Â°
= -ll-54c-2/2
=-^wÂ£> uÂ»>
where wl is the value of iv given by (111) ; that is,
u)jP=2 V41-30EwGK.
Then
and
y=_2xll-54?^1.3Ezrir
Knl
7"
Buckling of Deep Beams. 209
. . (120)
41-30- |3^Â£^â„¢C
â€” {-mfVI}- â€¢ â„¢
Consequently, W being the total load,
WZ2=m3Z8 v/4ErcCK
I 2 v/41-3 2 V Knj
= 12-85 VlSnCK- 23-1 |EG (122)
Thus the correction to WP due to the distribution of the
load along a line at height q above the centre line of the
beam instead of along the centre line itself is 23TyEC.
If the load were below the centre by an amount q the term
involving q would be added instead of subtracted, the beam
being in that case more stable than with the same load on
the central line.
The problem of the stability of a beam fixed at one end
-and free at the other was worked out in the first paper for
the following two cases : firstly, with a load P at the free
end and no other load ; secondly, with a uniform load per
foot and no load on the end. Now we will try to combine
these two loads.
Case 12. â€” Beam fixed at one end and free at the other,
and carrying a load P at the free end and a small uniformly
distributed load in addition. To find the condition for
instability.
It is assumed that the load on the end is much greater
than the uniformly distributed load.
Let w be the uniform load per unit length. Then the
bending moment G at distance x from the free end is
G = Pt? + ^2 (123)
.1 IV2.)'2
Therefore, neglecting - ___ compared with unit}*,
G2^PV + Pw3 (124)
Phil. Mao. S. 6. Vol. 39. No. 230. Feb. 1920. P
210 Dr. J. Preseott on the
Consequently the differential equation for the twist t is
d2r _ _ py+Piog3
EC
Let t = t1 + p, ....... (126)
where Tj is the solution of the equation
K"^=-EiTT" â€¢ â€¢ â€¢ â€¢ (127>
that is, rr is the value of t for the case where w = 0, which
is the problem solved in Case 3 in the first paper.
In the present case,
Kn { a? + 3? 1 = ~ KG 1 : + -f J r lTi + p I â€¢
Now w is a small quantity and p, being due to w, is
therefore also small. Then neglecting the product ivp
we get
By using equation (127) this last equation becomes
Â£{p+5?*>'}i . . (129)
that is, ^=-mV(p + rOTl), .... (130)
where 4 P2
EnCK'
(131)
r=%, (132)
P'
and, by equation (22) in the first paper,
Tl = 6{1"374 + 3.4.7.8""-r
The particular integral of (129) is
o= â€”rb
, ( mV 8mV
â€” vn J
(4.5 3.4.5.8.9
+ 3.4.5.7.^.9.12.13 * j ' 'lo3;
Buckling of Deep Beams. 211
The condition to be satisfied at the free end of the beam is
that the torque is zero, that is,
â€” =0, where # = 0.'
ax
This condition is satistied by the value of t in (126) if p
has the value given by (133).
The other condition is that
t=0 where x = L
Let
p â€” â€” rbxF(m2x2).
Then we have to solve the following equation for m2l2
bf(m2l2)-rblF(m2l2) = 0. . . . (134)
Let m2l2 = s-hv
where /(*)=0, (135)
that is, from equation (25),
5 = 4-012 (136)
Then f(s + v)-rlF{s + v)=0,
or f(s) + vf(s)-rlF{s) = 00 .... (137)
on neglecting v2. rv, and small quantities of higher orders.
Equations (135) and (137) give
f {*)
After the necessary arithmetic we arrive at the results
F(Â«)= 0-1014x5,
/v(5) = -0-0869x5.
Therefore v= -1-167 rl
â€” 1-1672?
ro
W
= -1-167â„¢ ,
ro
W being the total distributed load, and P0 the buckling
load when "W is zero.
i(> 2
212 Dr. J. Prescott on the
Finally,
w
mH2=s-l'167~,
or Fl2 W
â€” =4-012- 1-167^-
W
= 4-012 jl-O-291-jr-j
{1-0-291^}
Therefore p = 4-012 ^EnCK ^ n^W"
Z2
W
whence
= P0Jl-0.291p-}
= P0-0-291W, (138)
4-012
P + 0-291W = P0=^p v'EnCK (139)
The above is the equation that holds jnst when instability
occurs provided W is very small compared with P.
Now the buckling load when P is zero is, by Case 6,
Therefore the equation
P/2 W72 /^rw â€žAM
Â¥0T2 + 12*6 =VE,CK . . . (140)
is true in two cases :
(1) when P =0,
(2) when W = 0.
Moreover, this last equation can be written
P + 0-312 W = P0, (141)
which does not differ much .from (139). It seems very
probable then that equation (140) will be a good one for
all values of the ratio of W to P.
If the ratio between W and P is fixed it is possible to find
the actual values of these loads when buckling occurs, but
the problem is very awkward unless one of the loads is
small compared with the other. It is worth while, however,
to work out the case where P is small compared with W, so
as to see if equation (140) remains approximately true in
one more case. This we will now do.
Buckling of Deep Beams. 213
Case 13. â€” The same problem as Case 12 except that the
load P on the end is small compared with the uniformly
distributed load W.
Here, as in the last case,
G=Fx + iwx2.
\ wxj
= \w2xx ( 1 + â€” I approximately.
Therefore
dx2~ 4EnCKV WT
=-m6*i1+Sh â€¢ â€¢ â€¢ â€¢ (142>
^=4eSk 0*0
Now let t = t1H-/3, (144)
where (\ >Â»6^6 m12cÂ£12 ") ,-, , ^
Tl=a t^SJS+S. 6.11.12â€” â€¢)' â€¢ <145>
and therefore *H=_wVti (U6)
J.
Then equation (142) becomes
V *
4P
^2Ti d2p - .A 4P\ ,
= -mV{T1+â€” Tl + pJ,
neglecting the product Pp since both factors are small.
By means of equation (146) this last equation gives
fÂ£ + mVp=-^xhl. . . . (U7)
ax* r iv
Now if we differentiate through (146) we get
^-+m6A/=-4mVT1, . . . (148)
dr
t/ being written for â€” ^ .
p=
W
_Fdr1
id dx
T
= Ti +
F dTl
id dx
U14 Dr. J. Prescott on the
A comparison of (147) and (148) shows that a particular
integral of the former equation is
(149)
vu lv u< it-
Then it follows that
P rl-r.
(150)
is a solution of equation (142), and this solution satisfies
the condition that the torque is zero at the free end where
x â€” 0 ; that is,
â€”- â– = 0 where A' = 0.
dx
The only other condition that it is necessary to satisfy
is that
r = 0 where Â® = l,
that is,
Tt-i r^=0 where x=L . . . (151)
w dx
and from this equation m is to be found.
Let T^j\mx). (152)
p
Then, since m - is small,
w
f( mx + m - ) = f(mx) + m - f(mx)
P^Ti
10 tffo
(153)
It is now clear that equation (151) can be written in the
form
f(ml + m^)=0 (154)
But we found in the first paper (equation (55)) that the
solution of the equation
f(ml) = 0
was given by
m*P = 6-43 (155)
It therefore follows that the solution of equation (154) is
m^+?)3=6-43;
that is,
m3/3 ( 1 + yj ) â€” 6'43 approximately.
Buckling of Beep Beams. 215
Consequently
or W/2 P/2 â–
I^86+e9 = ^E"CK <156>
Now we see that, whether W is small compared with P,
or P small compared with W, the result expressed by (140)
is nearly true. Then it is sure to be nearly true for all
other positive values of the ratio W : P. The result may be
expressed roughly in the following form : â€” A load on the
free end of the clamped-free beam has approximately the
same effect in buckling as three times that load distributed
uniformly along the beam.
There is one assumption in the working of the last case
that should not be passed over without justifying it. It is
the assumption, made in the squaring of G, that P is small
compared with wx. This, of course, is quite true everywhere
except where x is small. But if we consider that the actual
term neglected, namely P2^2, is itself small in the region
where there is any possibility of error, and that the assump-
tion is wrong only over a very small range of values of ar,
it is clear that the error made is negligible..
There is still another point of view that will show the
justification for this assumption. The actual method of
solution consists in dropping a term from G2, and if we had
added a term of the same order we could have got
Then, by changing the variable to (x-\ ), our differential
equation would have reduced to the same form as in Case 6
where P was zero. It is easy to show by this method that
the solution of Case 6, with lx+ â€” J for x and (l+â€” \
for I, applies correctly to the present case provided powers
WlP
of beyond the first are neglected. But this is precisely
the solution we have obtained by taking a different value
of G2. Then it follows that the term neglected in G2 does
not affect the result to our degree of approximation.
Case 14. â€” Beam carrying a small uniformly distributed
load W and a much larger concentrated load P at the middle,
216 Dr. J. Prescott on th
and supported at the ends with just the necessary forces and
couples to keep the depth of the end sections upright.
This is a combination of Cases 4 and 7 in the first paper
with the condition that the ratio of W to P is small.
In this case, taking the origin at one end,
G = i(P + W^-i^a?2
W
= iQ^-.J-y.Â«2, ...... (157)
Q being written for (P-J-W).
Therefore
d2r _ QKv2 f 1 W;r i 2
it1" wl
dtf~ ""4EÂ«CKL ~QTJ T
^- I r approximately, (158)
where 4 Q
ni
4EhCK
159)
2W
If we also write r for the small quantity -7yr > then
Now let T = T! + />,
f mV
Tl==al*~47S+4.5.*.9
and therefore ^tj
fl+^V = mVT. .... (160)
where f m4<2?5 m8Â£c9 ") /1/>0>
j- â€¢ (lb^)
_2 +Â»/W1 = 0. . . . . (163)
Then (160) becomes, after making use of (163),
72
^ + m4^p = m4rlrr1, .... (164)
the product of the pair of small quantities r and p being
neglected.
A particular integral of (164) in series is
7>l4?* ft f _, ??24^4 ?7iV
r o . b L 4.9 4.8.y.
mi2xi2
13
4.8.9.12.13.T7+-} * ^165>
If we use this value of p in (161) we get a solution which
Buckling of Deep Beams. 217
satisfies the condition that t = 0 where x = 0, which is one oÂ£
the conditions of the problem.
Another condition is that the torque is zero at the middle.
This follows from symmetry, for clearly neither half can be
exerting a torque on the other. Then
dp + dp =0 when ^ â€ž ^ ^ ^ (166)
ax ax
that is, writing y for (^ml)*,
!u u2 u3 )
1-i + 4T5TÂ»-4.o.8.9.12+-j
rht (,, 10k 14wa
+ a^576r_479 + 4.8.9.13
-4.8.91lfl3.17+}=Â°- â€¢ â€¢ (167>
Now let M w2
/(") = 1-4 + 4.5.8â€” â€¢
4. 9^4. 8. y. 13 â€¢-'
then our equation is
/(u)+^F.(u)=0 (168)
Let us put w=5 f Â«,
where s is the solution when W = 0; that is,
/W = 0, (169)
and, by equation (33) in the first paper,
s=4-482 (170)
Then, retaining only the first powers of v and r and no
products, equation (168) becomes
or .Â» , rl
â€¢/(â€¢)+ go P(Â«)-0.
Therefore ,-l F(s) ,â€žâ€žâ€ž
218 Dr. J. Prescott on the
After the usual arithmetic we find that
F(Â» = 21-64,
/(Â»= -0-1974,
whence , 21*64 1fo97v2W
" = rl X 60 x 0-1974 = 1827X -Q-' ' { ]
and consequently
tlÂ»i4I4 = 5 + v = 4-482 + 1-827 ~
f 2W 1
= 4-482 < 1 -f 0407 x^ >,
Jm2Z2= a/4-482 j 1 + 0*407 ~ [> ,
M* = /4'4Â«2<1 + 0*407
or . QP i
wl
QI*
as
/ yy \ yy
Therefore, dividing by ( 1 + 0*407 â€”J and treating -tt
a small traction, \ . vc V
Wl
Z2CH 1-0-407 jj V =8 V'4'482 \/EnCK}
or QP-0-407 W/3= 16-94 ^/EMJK,
whence P/2 + ()-593W/2=16(J4 v'EnUET
This result can be written in the form
P72 W/2
1Â«VSWE"UK <"8>
Now when P is zero the present problem reduces to
Case 7, the solution of which is
28-31
But this differs very little from what we should get by
putting P = 0 in (173), which has been obtained on the
widely different assumption that W is small compared
with P. Then it is very likely that equation (173) is nearly
correct for all values of the ratio W : P.
The constants EC and K?i, which are involved in the
buckling loads, occur merely as constants in equations for
bending and torsion respectively ; that is, there is no
_
Buckling of Deep Beams. 219
assumption in this paper as to what Kn means except that
torque
Kn:
angle of twist per unit length
There is no assumption then that K has any particular form
such, for example, as is given by St. Venant/s theory of
torsion. The only assumption is that torque is proportional
to twist and that Kn is the constant expressing the quotient
obtained on dividing one by the other. Similar remarks
apply to EC, but there is no need to lay the same stress on
this as on the torsion coefficient because the bending
coefficient is better established, although even here G is not
rigorously the moment of inertia of the section if E is
Young's modulus. The point of the preceding argument
is this, that EC and Kn are a pair of coefficients which
should be obtained by experiment for any particular beam
that is to be used for testing the theory of this paper. It
would not be right to calculate EC and Kn, even if E and n
were known accurately, for that would be burdening the
theory of buckling beams with whatever errors are contained
in the theories of bending and of torsion. Of course, when
the buckling formulae have to be used in practice it will
usually be necessary to be content with calculated values of
C and K and assumed values of E and n, for that will be
the best that can be done. But, in the testing of the results
in the experiments carried out by Mr. Carrington, EC and
Kn are found experimentally and their values substituted in
the expressions for buckling loads, and these are compared
with the actual experimental buckling loads.
In using the preceding results in practice, where it will
usually be necessary to calculate everything, the value of C
is the same as in the bending of beams and the best value
of K is the one given by St. Venant's theory of torsion.
For a beam of rectangular section, breadth b and depth d,
these values are
c=Ty,v,
K=Â£W(l-0-630jV
the latter being the approximate value of St. Venant's
torsion coefficient when b is less than ^d.
220
Mr. H. Carrington on tlu
Appendix by H. Carrington.
Experiments were performed to determine the degree of
accuracy with which the buckling loads for cantilevers
loaded at one end, as calculated by the mathematical ex-
pression in (1) below, agreed with those obtained by
experiment. Five steel strips were used, each of which was
straight, free from dents, and accurately ground.
The expression for the buckling load is
P =
4-012 VEttCK
I2
(1)
where I is the length under test, EC the least flexural
rigidity, and Kn the torsional rigidity, so that in order to
calculate P it was first necessary to determine these
rigidities.
The flexural rigidity of a strip was obtained by arranging
it as a cantilever and loading it with weights suspended
from the free end by a fine wire. The deflexions (S) of the
free end corresponding with increasing increments of load
were noted and plotted one against the other. The lines so
obtained are given in fig. 1, and the flexural rigidity (EC) is
Fie. J.
Deflection ( I division = Ol inch)
WP
w
given in each case by the expression EC= -^ ^ , where
do o
is the slope of the line and I is the length under test. The
thicknesses of the strips (see table below) were small
compared with the lengths under test (10 in. and over) and
the deflexions due to shear were accordingly negligible.
In order to obtain the torsional rigidities the strips were
Budding of Deep Beams. 221
fixed in a delicate torsiometer which was capable of trans-
mitting small torques of known amount, about the longi-
tudinal axes of the specimens. The angles of twist corre-
sponding with different gradations of torque were noted and
plotted one against the other, and the curves are reproduced
T
in fig. 2. The torsional rigidity Kn is given by Kn= ^ Z,
T
where -pr is the slope of the curve and I is the length of the
6
corresponding strip under test.
Fisr. 2.
Angles of ti/s/st - (l - i0n radian)
The values of the rigidities for each of the five strips are
given in the table (below) and also values of P calculated
by the expression (1) and corresponding with two lengths
for each strip. It may be noted that no dimensions of the
strips other than the lengths under test are involved in
equation (1).
The buckling loads were obtained experimentally as
follows : â€” each strip was firmly gripped in a vice and pro-
jected horizontally outwards with its sides vertical. The
load was transmitted by weights suspended by a fine wire
which passed through a small hole drilled through the mid-
depth of the specimen near its end. When the buckling
load was almost reached the specimen could vibrate slowly
from side to side and finally came to rest in the central
position. When the buckling load was just past, the
222
The Buckling of Deep Beams.
o
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o
LO
Â©
03
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j
OJ
C
c
X)
u
i
Â£
, â€”
be
g S W
5
o
c
>
2
* ^
bfi Â«*â–
,b0
c
-^
X
-3
0 râ€”
? 9
*
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sâ€”
" i '&
U
Si
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03
On a Simple Property of a Refracted Ray. 223
specimen could still vibrate slowly from side to side, but
came to rest in a position displaced on one side or the other
of the central position. The buckling load was taken as
that which would cause the beam to come to rest so that the
displacement from the central position was the smallest
appreciable. The results of all the experiments can be seen
at a glance from the accompanying table.
It was found of importance that the sides of the strips
should be exactly vertical, for if they were very slightly
inclined , a displacement was noted at comparatively small
loads. A delicate adjustment was necessary, so that when
the buckling load was reached the strips would remain at
rest, very slightly displaced, on either one side or the other
of the central position.
In every case the buckling loads were determined experi-
mentally before their values were calculated by equation (1)
in order that the determination of the experimental loads
should not be influenced by a knowledge of the calculated
values. This was necessary because it was difficult to decide,
within 2 or 3 per cent., at what load buckling began.
XX. Sote on a Simple Property of a Refracted Ray.
By Alice Everett *.
LET AP be a ray incident at a point P on any refracting
surface, BP the refracted ray, CPN the normal, C the
centre of curvature in the plane of incidence ; <Â£, ' + ^r = fy, say. The
angle 7 plays a leading part in some modern optical
generalized formulae (see Optical Society's Transactions,,
vol. xx. pp. 23-31, Nov. 1918, where it is pointed out that,
for a spherical refracting surface, 7 is also the angle made
with the axis by the line joining C to the points of inter-
section of the rays with the aplanatic surfaces), but objection
has been raised that this angle is not readily visualized.
The following simple property which, curiously enough,
seems hitherto to have escaped notice, may be of service
in this respect.
* Communicated by the Author. From a communication to the
Transactions of the Optical Society for March 1919, p. 203.
224 On a Simple Property of a Refracted Ray.
If PT be the tangent at P to the circle circumscribing
the triangle APB, then the angle TPN, between this
tangent and the normal produced, is equal to -\-yjr' =7,
since the angle APT between this tangent and the chord
AP is equal to the angle at B in the opposite segment
of the circle, and the angle APN is 0. Without actually
drawing the circle APB, the direction of the tangent can
usually be judged by inspection. If now the transversal
rotate about C, while the rays remain fixed, the tangent
will rotate about P at the same rate, but in the opposite
direction, the angle swept through in either case being
A\jr = increment of ijr = increment of t\fr'. When A-v/r attains
the value it â€” 7, the tangent to the circle coincides with the
normal P(1 to the refracting surface, and if CB'A' be
the corresponding position of the transversal, then
zBCB' = 7T-7, z.PB'C = /dit* + ]P$=09 .... (2)
k being equal to n/a. The solution of (1) appropriate to
our purpose is
J> __ e%(nt+1cy sin 8) f J^gikx cos 8 _j_ Iggâ€”ikx cos 8\ /g\
the first term representing the incident wave travelling
towards â€” #, and the second the reflected wave. From (3)
we obtain for the velocity u parallel to #, and the con-
densation 5, when A' = 0,
M=^=^^+^8in0)^cos^(A-B), ... (4)
(XX
1 dcf) in
a dt a
a*=-^^=-â„¢Â«Â«*+*rinÂ«(A4-B), . . (5)
so that u .B- A ,n,
as = 00sdBTA (6>
For the motion inside a channel we introduce in (1) on
the left a term hdcf>/dt, h being positive, to represent the
dissipation. Thus, if <Â£ be still proportional to eint, we have
in place of (2)
d2(f>ldx2 + d2(f>/dy2 + d2/dz2 + ]e'2cl> = 0, . . (7)
where k'2 is now complex, being given by
J,'2==k2_inh/a2 (8)
If we write k' = k1 â€” ik2, where &1? k2 are real and positive,
we have
k]2-k22 = P} kjc^nh/a* (9)
At a very short distance from the mouth of the channel
d2(j>/dy2, d2(p/dz2 in (7) may be neglected, and thus
<Â£ = ^{A'cos*^ + B'sin&'#}. ... (10)
Reflexion of Sound from a Perforated Wall. 227
If the channel be closed at #= â€” I,
A' sin k'l + B' cos 1/1=0,
and we may take
4> = A"cosk'(x + l)eint (11)
From (11) when x is very small,
u = dcfy/dx =-/â€žâ€¢' A" sin kfl . ein\ . . . (12)
as=-a-1d/dt=â€”ikA." cos k'l.e**, . . (13)
â€¢so that u 1/ ,
= ry tan kl (14)
as ik
Now, under the conditions supposed, where the transition
from the state of things outside to that inside, at a distance
from the mouth large compared with the diameter of a
channel, occupies a space which is small compared with the
wave-length, we may assume that s is the same in (6) and
(14), and that
(3 tan kx : % tan kx -| : v . (23)
The denominator of (22) is obtained (with altered sign)
by writing â€” S for S in (23).
In what follows we are concerned with the modulus of B.
Leaving out factors common to the numerator and denomi-
nator, we may take
Mod.2 Numerator = { jfeg- *lta"^2-Â£2 tan k, ]â– '
' ( /j^tanikc, 7\ 7 &2 tan ik0 ) 2 _
+ \ rs ~i ~ki ) tan ** + i â€” : } â€¢ (u>
Reflexion of Sound from a Perforated Wall. 229
The evanescence of B requires that of both the squares in
(24), or that
Â£S = -^â€” - â€” 2 + k2 tan l\ = ik1 cot ik2â€”k% cot fcly (25)
or again with elimination of S,
z'/^tan ik2 -f- cot zÂ£2) = k2 (tan A^ -f cot ki) ,
whence
*1sin2*1 + Â«ife2sin2tifc2=0, .... (26)
or in the notation of the hyperbolic sine
kt sin 2&1=&2sinh 2k2 (27)
If this equation, independent of cr, a ', and cos 6/, can be
satisfied, it allows us to find kY from an assumed k2, or con-
versely, and thence k by means of (9).
The next step is to calculate S by means of one of equations
(25). If S, so found, > cos 0, we may choose a' /a so that
B shall vanish ; but if S < cos #, no ratio a' fa will serve to
annul the reflexion. If the incidence be perpendicular,
S must exceed unity. If S were negative, the reflexion
would be finite, whatever may be the angle of incidence and
the ratio y and
increases with great rapidity. On the other hand, y vanishes
whenever kx is a multiple of \tt, although the successive loops
Reflexion of Sound from a Perforated Mall. 231
increase in amplitude in virtue of the factor kx. The solutions
of (27) correspond, of course, to the equality of the ordinates
y and y'. It is evident that there are no solutions when y is
negative. The most important occur when k2 is small and
2kx just short of it. But to the same small values of k2
correspond also values of 2/(1 which fall just short of 37r,
D7T, &c, or which just exceed 2-7T, 4-7T, &c. More approxi-
mately these are
4 cos mir.k22 ,
2ki = 7mr-\ , .... (32)
where m = l, 2, 3, &c.
In order to examine whether these solutions are really
available, we must calculate S. By (25)
ks=kJi-lkA(mJr + 2cosm,r-^
\ o /\ 2 mir J
T . / mir 2 cos mir . k22\
+ k2 tan I -~- H .
\ 2 mir /
If m is odd, we have approximately
*S-jg(l+V); (33)
and if m is even,
a-^ii+V^-i)}. â€¢ â€¢ (34)
Since Â£ is approximately ^??i7r, we see that when m is odd,
S is large, and the condition of no reflexion can be satisfied,
as when m = l. On the other hand, when m is even, S is
small, and here also the condition of no reflexion can be
satisfied, at any rate at high angles of incidence.
It should be remarked that high values of m, leading to
high values of k, correspond with overtones of the resonating
channels.
A glance at fig. 1 shows that there is no limitation upon
the values of the positive quantities kl and k2. And since kk
is always greater than k2, k, as derived from k\ and k2, is
always real and positive.
So far we have supposed that the values of /cb corresponding
with small values of k2, are finite, as when m = l, 2, 3, &c.
But the figure shows that solutions of (27) may exist when A:^
as well as k2, is small. In this case we obtain from (31)
ft^va+jv), (35)
making W^kf-k^U^ (36)
232 Reflexion of Sound from a Perforated Wall.
Hence by (24)
JfeS = kx tanh &2 + k2 tan ^ = 2Â£22(1 + p22), . (37)
and S=v/3.(l + p22) (38)
Here again the condition of no reflexion can be satisfied,
whatever the angle (6) of incidence, by a suitable choice of
(7l/(T. But the damping is no longer small, in spite of the
smallness of k2, since k2 is not now small in comparison with
k} and k. On the contrary, kx and k2 are nearly equal, and B
is small in comparison with k2, so that this case stands apart.
Not only is it always possible to find a series of values
of ki satisfying (27) with any assumed value of k2, but the
values so obtained make S positive. For in (25) k^ k2, tanh k2
are positive, and so also is tan An, since
tan &!= sin 2 kj 2 cos2 k1}
and sin 2kl is positive.
It is a question of some importance to consider whether
when a, o\ and 0, determining S, are given, the reflexion
can always be annulled by a suitable choice of kY and k2.
It appears that the answer is in the affirmative. Let us
consider the various loops of fig. 1 which give possible
values of k2. The ranges for 2k\ are from 0 to tt, from 27r
to 37r, from 4-7T to 57r, and so on. As we have seen, the
intermediate ranges are excluded. In the first range between
0 and 7r we found that S may be made as great as we please
by a sufficiently close approach to it. At the other end
where #i = 0, the value of S was \/%, or 1'7320. This is the
smallest value which occurs. When 2#1 = ^7r, it appears
that k2 â€” '5656, Â£ = '5449, and S = 1-947. And again, when
2A;1 = f7r, #2 = *5795, S = 1'964. We conclude that within
this range some value of k\ with its accompanying k2 can be
found which shall annul the reflexion, provided S exceed
1*7320, but not otherwise.
In each of the other admissible ranges, S takes all positive
values from 0 to go . At the beginning of a range when 2&a
slightly exceeds 27r, 47t, &c, S starts from 0, as appears
from (34) ; and at the end of a range, as 37r, 57r, &c. are
approached, S is very great (33). Within each of these
ranges it is possible to annul the reflexion by a suitable
choice of ku k2, whatever cr, o\ and 6 may be.
If the actual value of S differs from that calculated, the
reflexion is finite, and we may ask what it then becomes.
If we denote the value of S, as calculated from Â£l5 k2, by S0,
(24) gives
Mod.2 Numerator = P(S-S0)2{l + tan2 hx tanh2 k2],
The Quantum Theory of Electric Discharge. 233
and in like manner (by changing the sign of S) ,
Mod.2 Denominator=A:2(S + So)2{l+tan2fitanh2A:2} ;
and hence
Mod.*B=(Â§^|)2, (39)
where S = cos'0(â€” 8
only about 2250Â° C. These figures emphasize the great
importance of the protecting layer of carborundum, and it
deserves to be mentioned that some of the most important
results achieved in the course of the present investigation
were directly due to this precaution.
Â§ 4. Luminous phenomena observed in the vicinity of the
electrically heated plate of graphite.
Observations were made by forming an image (about twice
actual size) of the heated plate upon a white cardboard screen.
This plan proved not only very convenient, but it also had
the further advantage over direct observation through dark
glasses, of showing the phenomena in their true natural
colours, â€” a most helpful adjunct for interpreting the meaning,
of the luminous effects displayed.
As the temperature of the plate is gradually raised,,
yellowish vapours begin to form along the undersurface
and a continuous stream of similar vapours is seen to rise
upwards from above the plate. At temperatures of from
2300Â° to 2500Â° C, the coloration of the vapours beneath the
plate changes to bluish grey and, furthermore, they present
now a, sharply-defined outline as sketched in fig. 3. In the
region above the plate, where the temperature is much lower,,
the colour of the rising vapours continues yellow, with here
and there a greyish streamer or patch. In the neighbour-
hood of the plate, principally just above it, are also seen
red-coloured regions. All these colour effects are due, no
doubt, to the various light radiations emitted under the pre-
vailing temperature conditions, by the vapours driven out
from the graphite plate and the carborundum.
In order to account for the sharp demarcation of the
incandescent vapours beneath the plate two factors must be
taken into consideration â€” namely, the continuous effusion,
from the undersurface, of vapour which is being forced
downwards into the protected space, and the upward
draught of air, as explained in Â§ 2. At the boundary sur-
face of the protected space the hot vapour comes into contact
with the air current, is cooled with consequent changes in
its radiating properties, and is then immediately carried
away upwards. Since it is reasonable to assume that, whilst
the temperature of the plate remains constant, both the
quantity of vapour passing through the protected space and
Titanium, and Vanadium by Thermelectronic Currents. 251
the velocity of the upmshing air will not vary appreciably,
we may expect that some stable regime will become estab-
lished in which there are no rapid variations in the position
Fig-. 3.
MM #
Rising vapours
Carborundum
powder
Graphite
plate
B!u/sn vapours
Plate temperature : 2300-2700Â° C.
of the boundary surface between the vapours and the sur-
rounding air currents. The sharply defined outline of the
bluish vapour might accordingly be caused by a steady
state of the acting forces and by the continual clearing
away of the superfluous vapours through the upward rush
of air.
Although the existence of appreciable ionization currents
through the luminous vapours, under these temperature con-
ditions, was easily shown by means of a pair of exploring
electrodes, the spectroscopic results did not, however, indi-
cate that the light radiations emitted by these vapours were
entirely governed by them. A most remarkable effect, both
luminous and spectroscopic, was however observed when
the graphite plate was raised to a temperature of about
3000Â° C. In immediate contact with the undersurface of
the hot plate and suspended from it as it were, there ap-
peared a sharply defined luminous band of pinkish hue,
stretching right across the space between the clamping bars
252 Mr. G. A. Hemsalecli : Excitation of Spectra of Carbon,
and extending downwards to a distance of from 1 to 2 mm.
(fig. .4). For convenience' sake, this luminous band will
henceforth always be referred to simply as the red fringe,
although there is a distinct violet shade in its colour.
Fi-. 4,
+
Red -fringe
Rising vapours
Carborundum
powder^
Graphite
plate
Bluish vapours
Relative position of
spectrograph slit.
Plate Temperature : 3000Â° C,
A preliminary examination of the spectrum of the red
fringe, which was found to be entirely different both as to
character and composition from that given by the luminous
vapours situated beneath the fringe, at once suggested
thermo-electrical excitation as the cause of its emission.
It was therefore provisionally assumed as a working hypo-
thesis that, owing to the very great degree of ionization
prevailing at this high temperature, part of the heating
current had passed out of the graphite plate into the space
below and that its path was revealed by the formation of the
red fringe.
Titanium; and Vanadium by Thermelectronic Currents. 253
Â§ 5. Influence of a transverse magnetic field upon the
visibility of the red fringe.
If, as has been assumed, the red fringe were caused by the
passage of an electric current outside the plate, it should be
acted upon by a transverse magnetic field. In order to test
this graphite plates were mounted between the hollow field
coils of an electromagnet in such a way, that the heating-
current flowed at rio'ht angles to the direction of the lines of
magnetic force. The temperature of the plate was raised to
about 2700Â° C, at which stage the red fringe is ordinarily
not yet visible. As soon, however, as a magnetic field of
from 125 to 175 C.G.s. units was put on with the force acting
upon the heating current in a downward sense, the red fringe
immediately appeared. On taking the magnetic field off, the
red fringe disappeared. On reversing the direction of the
magnetic field, no red fringe was observed. One would
have expected, in this latter case^, to see a red fringe appear
above the plate, but undoubtedly, owing to the prevailing-
convection currents, the vapours do not become sufficiently
ionized under these temperature conditions to allow of the
passage of an electric current. But when the plate was
raised to a temperature of about 3000Â° C, so that the. red
fringe was well visible beneath the plate even without the
aid of a magnetic field, then the application of an upward
acting magnetic force of 1300 C.G.s. units brought out a red
fringe above the plate, whereas the fringe beneath the plate
practically disappeared. On taking the field off again the
red fringe above disappeared, whereas at the same time that
beneath the plate reappeared. By applying Fleming's hand-
rule it was easy to show that the displacements of the red
fringe were quire in accordance with the laws of electro-
magnetic induction, which is a direct proof that the red
fringe is governed by the flow of an electric current.
Furthermore, the appearance of the red fringe cannot be
governed by temperature alone, for otherwise it would mean
that when a magnetic field of from 125 to 175 units acts in
the downward sense upon the plate at 2700Â° C. its tempera-
ture should increase by about 300Â° C, which, as the results
of my temperature determination (Â§ 3) clearly show, is not
the case.
But, in addition to these facts and arguments in favour of
the electrical nature of the red fringe, a most important
factor, which also gives strong support to this view, is the
following observation made when the red fringe was acted
upon by a transverse magnetic field in a downward sense : â€”
luminous streamers were seen to pass out of the fringe into
254 Mr. G. A. Hemsalech : Excitation of Spectra of Carbon,
the space below, forming bright and most sharply defined
spiral or other paths, such as might be expected to result
from the action of a magnetic field upon a stream of
luminous particles carrying electric charges.
Â§ 6. Origin of the red fringe.
It may be useful, at this stage, to briefly inquire into the
nature of the vehicles which convey the electric current in
the red fringe. It is well known from the work of many
physicists, in particular from Professor Richardson's extensive
researches, that at temperatures much below that at which
the red fringe is formed, electric currents are passing
through the ionized vapours or gases in the neighbourhood
of an electrically heated carbon rod or metal wire. It is
generally assumed that these currents are caused by the dis-
placement of ions under the influence of the acting electric
field, and Professor Richardson has therefore proposed to call
them thermionic currents. There is no doubt that the
electric currents, which I was able to register between two
exploring electrodes held in the luminous vapour beneath
the heated plate, at the lower temperatures (see Â§ 4) were
of this nature. Now, these currents do not seem to have
any appreciable influence upon the character of the spectra
emitted by the vapours, which, as will be shown later, are
practically identical with those given by these same vapours
in the outer mantles of flames. But also in other respects
the thermionic currents differ greatly from the current
which causes the red fringe â€” namely in a transverse
magnetic field they are much less acted upon than the
latter. Whereas the red fringe is extremely sensitive to
the magnetic force and a field of a very few units suffices
to produce a distinct spiral or helical path, a field of at least
500 units is necessary to form an ill-defined and relatively
undeveloped curved path in the luminous vapour with ther-
mionic currents. It seems to me that this difference in
behaviour between the thermionic current and that causing
the red fringe might possibly be accounted for by assuming
that the particles which convey the electric current are of
different masses in the two cases. Thus, whereas the ther-
mionic current is probably due to the motion of relatively
heavy particles, in this case perhaps carbide molecules, the
current in the red fringe would arise from the displacement
of relatively light particles. These are perhaps constituted
of free radiating atom-ions formed by the electrons which
emerge in large quantities from the hot graphite plate, and
the paths of which are under the control of the magnetic
Titanium, and Vanadium by Thermelectronic Currents. 255
field set up by the heating current, as explained in Â§ 8. The
origin of the red fringe emission would thus be, in some way,
connected with the process of generation of these atom-ions.
As it will be necessary, especially for spectroscopic purposes,
to make a clear distinction between the thermionic current
and that producing the red fringe, I propose to call the
latter the thermelectronic current.
Â§ 7. Spectroscopic analysis of red fringe and luminous
vapours.
A two-fold magnified image oÂ£ the incandescent graphite
plate and the luminous phenomena below it was accurately
i'ocussed on the slit of the spectrograph, already described in
a previous communication *. The image was adjusted in
such a way that the slit passed through the middle part of
the plate and perpendicularly to it, as indicated by the
dotted line on fig. 4. By this means the spectral changes
occurring along the distance from the hottest part in the
immediate vicinity of the plate, down to the region of com-
plete extinction of the luminous vibrations, could be observed
at a glance or recorded photographically. The times of
exposure for the photographic records varied from 3 seconds
at the highest to over half a minute at lower temperatures.
The spectrum of the vapours which form beneath the plate
is of course due to the elements contained in the graphite
and carborundum as impurities, probably in combination
with carbon as carbides. At lower temperatures appear
the lines of Na, K, Li, Sr, Ca, Mn, Al, and Fe, and the
whole spectrum is observed to grow progressively .in in-
tensity and development as the plate temperature gradually
rises. In addition to the line spectrum there is also seen
a continuous spectrum which is particularly strong near
the lower edge of the bluish vapour and extends to the
same distance downward as the latter. Now, the lines
emitted by the various impurities expelled from the graphite
and carborundum generally pass well below the border of
the continuous spectrum ; this indicates that their emission
centres travel to a greater distance from the plate than the
bulk of the bluish vapour. Hence there appears to be no
direct connexion between the line emission observed and the
sharply bordered cloud of bluish vapour described in Â§ 4.
It is possible that the bluish vapour is formed only along the
boundary surface of the protected space, constituting as it
were a kind of envelope, and that the emission centres, to
* Heinsalech, Phil. Mag. vol. xxxiii. p. 7 (1917).
256 Mr. Gr. A. Hemsalech : Excitation of Spectra of Carbo?i,
whicli are due the impurity lines observed under these con-
ditions, are located within the enclosed space â€” namely, in
the hot region extending down from the central part oÂ£ the
plate. In order to account for the greater extension down-
wards of the line emission as compared with the continuous
emission due to the bluish vapour, we may suppose that the
emission centres, which are probably constituted oÂ£ charged
particles (perhaps carbide molecules, see Â§ 6), travel along
lines of force under the action of the electric field established
along the graphite plate by the passage of the heating current.
As it will be of advantage for the sake of clearness to desig-
nate the particular vapour which gives out the line emission,
as distinct from the bluish vapour, it will always be referred
to as the luminous vapour.
With regard to the iron lines it will be shown hereafter
that their relative development is in accordance with that,
which I have previously observed for the same element in
flames and in the tube-furnace. From this we may conclude
that, also in the present case, their emission is caused by
thermo-chemical excitation â€” namely, the action of heat on a
chemical compound of iron, in this case probably a carbide.
A characteristic feature of all the lines emitted by the
luminous vapour is that they die out gradually on passing
downwards. This is, of course, to be expected, if these
radiations are, as I presume, controlled by the plate tem-
perature, for the latter naturally decreases with increase of
distance from the plate. In this respect these lines behave
very similarly to what they do in flames, in the mantles of
which they are likewise observed to die out only gradually
on passing into cooler regions.
When the plate temperature is raised to about 2700Â° C.
a new spectrum begins to develop in the immediate vicinity
of the graphite plate, with the appearance of the carbon*
bands at 3883 and 4216 and of numerous lines due to
titanium and vanadium. As the temperature is gradually
increased this spectrum gains in prominence and more
earbon bands appear in the visible part until nearly all the
bands of the Swan spectrum are out. Finally, in the red
part is seen a group of most intense, hazjr, and broad bands,
which when the spectrum is fully developed constitute its
most brilliant feature. Now the very striking and dis-
tinguishing character of this spectrum is, that the lines
* Hunge and Grotrian have recently concluded that these bands are
due to nitrogen. As a result of my own experiments, "which were like-
wise made at atmospheric pressure and of which I hope to give an
account on a future occasion, I am unable to endorse their view. Nor
do my experiments indicate that the presence of N is always essential
for their emission.
Titanium, and Vanadium by Thermelectronic Currents. 257
and bands which compose it, unlike those emitted by tKe
luminous vapours previously described, pass only a short
distance down from the graphite plate and stop quite
abruptly, as though the exciting agent had suddenly ceased
to act. Evidently then, this spectrum cannot be entirely
controlled by the plate temperature, as otherwise its lines
and bands, like those of the luminous vapour emission,
would die out only gradually. We must therefore trace
its origin to some other cause, and it is only natural to
connect the emission of this spectrum with the formation of
the red fringe. A direct experimental proof for the reality
of this connexion will be given in Â§ 9. Thus, the emission of
the carbon bands and of the lines of titanium and vanadium
is in some way caused by the thermelectronic current which,
at these high temperatures, passes through the strongly
ionized vapours in the immediate vicinity of the graphite
plate. The emission would therefore be due to what I have
previously called thermo- electrical excitation *, and it is no
doubt of the same nature as that which I had preconceived
to exist in a high temperature tube resistance-furnace.
The spectrum of the red fringe and of the luminous
vapours are reproduced on Plate II. The narrow strip of
sharply defined and strong continuous ground, which forms
the upper edge of each section, is due to the luminous
emission by the exposed edge of the incandescent graphite
plate. The ultra-violet end a was obtained with the special
furnace arrangement in which the vapours beneath the plate
are better protected from air currents, and sections b and c
with the clamping bars described in Â§ 2. In this latter case
the vapours are more exposed to air draughts and therefore
the spectrum lines# do not pass down quite so far. In the
zone of the red fringe the two emissions are naturally seen
in superposition, but by reason of the abrupt extinction of
the lines and bands which compose the red fringe spectrum,
the separate existence of each emission is clearly brought out.
Some of the low temperature lines are seen in absorption upon
the continuous spectrum due to the edge of the graphite
plate. This is of course caused by the constant stream of
excess vapour which is being carried upwards in front of the
plate by the aforesaid air currents. It is also well to draw
attention to the behaviour of the H and K lines of calcium.
These lines are seen to pass down a much shorter distance
from the plate than neighbouring iron lines of equal or
even less intensity, such as the group at 3920. They are
* Hemsalech, Phil. Mag. xxxvi. p. 295 (1918).
Phil. Mag. S. 6. Vol. 39. No. 231. March 1920. S
258 Mr. G. A . Hemsalech : Excitation of Spectra of Carbon,
therefore not emitted at so low a temperature as iron lines.
This fact is indeed quite in harmony with M. de Watte ville's
and my observations regarding the flame spectra of these
elements. In the mantle of the air-coal gas flame the iron lines
are well developed, whereas no trace of H and K is- seen.
But the latter are quite intense in the oxy-coal gas flame at
2400Â° C. It would therefore seem that also in the present
case the H and K radiations are caused principally by thermo-
chemical excitation, and they do not appear to be appreciably
affected by the thermelectronic current.
The following tables contain all the lines and bands of the
spectrum of the red fringe, but only those of the luminous
vapour which were observed by me likewise in a carbon tube
resistance-furnace. Lines to which an asterisk is affixed
are marked on the plates to the nearest Angstrom unit.
Titanium.
Relative
intensities.
Relative
intensities.
A
r
^
~>
*.
Red fringe.
Arc.
Spark.
X
Red fringe
Arc.
Spark.
3635-47
000
15
3
4078-47
000
8
4
3642-68
000
15
3
4301-08
1
15
3
3653-49
00
15
4
4305-91
1
20
8
3671-66
000
4
3
4314-80
000
5
3
3689-89
000
3
2
4393-93
00
5
2
3717-39
000
5
2
4512-74
00
15
4
3729-77
1
8
4
4518-03
00
15
4
*3741- 14
1
15
2
4522-80
00
15
4
*3752-87
1
15
5
4527-31
00
15
4
3753-63
*3771-64
00
JL
3
4
3
3
*4533-25
*D34-78
1
20
15
5
4
3900-53
00
5
50
â€ž4 35-92
0
8
3
3904-77
0
10
5
.M6-00
20
4
3924-52
1
8
3
4544-70
0
10
3
3929-87
1
8
3
4548-77
00
8
3
3947-75
1
10
3
4656-47
0
8
3
3948-66
14
12
4
4667-59
0
10
5
3956-28
14
15
4
4681-91
0
20
10
395821
2
15
5
4991-08
1
20
10
3962-86
0
8
3
4999-51
1
20
10
3964-27
J_
8
3
5007-22
1
2
20
10
*3981-77
1
15
3
*5014 26
1
20
8
3982-54
0
8
3
5039-96
1
10
3
*3989-77
0
20
6
5064-66
1
10
4
*3998-85
0
20
6
5173-74
0
15
5
*4009-14
1
8
4
519297
0
20
8
400968
000
4
2
*52l0-39
1
20
10
*4024-57
1
2
10
3
Titanium, and Vanadium by Thermelectronic Currents. 259
Wave-lengths are given in international units and the
scale of intensities is that outlined in a former commu-
nication *.
Some lines situated in the bright tails of the carbon bands
have probably escaped attention. The relative intensities in
arc and spark were obtained from the observations of Exner
and Haschek. A comparison of these results shows that the
red fringe spectrum of titanium is composed of the brighter
arc lines, and only one so-called enhanced line has been
recorded â€” namely, X 390053. As to the character and
origin of this spectrum very little can at present be said or
suggested, because the flame spectra of this element (chemical
and thermo-chemical excitation) are as yet practically un-
known. It is, however, of interest to mention that none of
these lines were observed by me in an ordinary arc between
electrodes made of the same kind of graphite ; but they ap-
peared near the cathode in heavy current arcs of from 20 to
#0 amperes. No trace of them was observed in the ordinary
condensed spark between graphite poles, but a few of the
brighter ones were detected in the self-induction spark. It
is therefore remarkable that they should constitute such an
important part of the red fringe spectrum.
Vanadium.
Relative intensities.
*
Relative
intensities.
X.
r
Eed fringe.
Arc.
Spark.
X.
Red fringe.
Arc. Spark.
3902-25
0
4
2
4128*10
h
10 10
409268
00
15
3
4131-98
JL
10 10
4099-80
00
20
2
413447
000
10 10
4105-20
00
10
4
*4379 24
1
30 R 30 R
4109-78
000
15
10
4384-73
1
30 30 R
4111-80
1_
30 R
2
4389-98
6
20 20 R
4115-17
00
5R
2
*4408-50
i
10 15 R
4116-50
000
15
5
4586-37
00
10 8
4123-65
000
5
3
4594-09
00
10 10
This spectrum is likewise composed of the brighter arc
lines only. It is of course possible and even probable, that
some lines have been overlooked, as in the case of titanium ;
this is, however, inevitable on account of the presence of a
strong continuous ground and of the carbon bands, which
fact, coupled with the relatively low dispersion of my
* Hemsalech, Phil. Mag. vol. xxxiii. p, 9 (1917).
S2
260 Mr. G. A. Hemsalech : Excitation of Spectra of Carbon,
spectrograph, would naturally tend to mask or obliterate
some of the lines.
Also in this case it is impossible to form an adequate
opinion as to the true character of the spectrum emitted
by vanadium, on account of the absolute lack of information
with regard to its flame spectra.
Carbon.
Bed edge of band.
\. Relative Intensity.
#3861-71
4
1
1
#3871-39
6
\
*3883'40
10
j
So-called
#4167-61
#4180-82
1
1|
i
> Cyanogen
bands.
#4197-08
2
#421596
3
i
j
J
4684-76
00
i
1
4697-39
4715-13
.1
2
i
Gi
â€¢oup
IV.
#4737-00
3
j
Swan
Spectrum.
Â«
* 51 29-24
*5165-12
1
4
l
j
Gi
-oup
III.
5585-28
'â€¢5635-21
0
1
}
Gi
â€¢oup
II.
_>
The spectrum is composed of the so-called cyanogen bands
and three groups of the Swan bands. I have no doubt that
group I. of the last-named bands â€” namely, that with head
at \ 6188, would likewise show were it not for the presence,
in this part of the spectrum, of one of the hazy broad red
bands. A higher dispersion will probably reveal it. The
bands at 3883 and 4216 are particularly intense and,
next to the hazy red bands, they constitute the most pro-
minent feature of the red fringe spectrum ; also I feel
inclined to attribute the violet tint in the red coloration of
the fringe to the presence of these bands.
Although there is nothing noteworthy in the general
aspect of the carbon bands as excited by the therm-
electronic current, it is however of prime importance to
point out the marked fundamental difference which dis-
tinguishes this mode of their excitation from that which
determines the emission of these bands in the explosion
Titanium, and Vanadiumby Therm-electronic 'Currents. 261
regions of various hydrocarbon flames. As was already
stated at the beginning of this paragraph, the first band to
appear under red fringe conditions is 3883. As the intensity
of thermo-electrical excitation increases, the various groups
of bauds appear in the order as given in the table â€” namely,
first the so-called cyanogen bands and finally those of the
Swan spectrum. Now this order of development of the
carbon spectrum is reversed when chemical excitation is the
cause of their emission. With the feebler chemical actions
in the air-coal gas cone the Swan bands appear indeed alone,
and it is only in the cone of the oxy-acetylene flame, where
chemical actions of a very violent nature prevail, that the
bands at 3883 and 4216 are likewise brought out. It is also
well to mention the fact that the carbon bands are not
emitted by the hottest parts of these flames â€” namely, the
region in the mantle just above the cone ; hence their emis-
sion cannot be caused solely by thermal or thermo-chemical
actions, and this conclusion agrees well with the results of
the present experiments.
Red bands of unknown origin.
X. Relative intensity.
6059 1
*6176 2
*6313 2
*Â£Â±$2 1
As these bands have no well-developed edges the settings
were made upon their middles, as nearly as was possible to
judge. The numbers representing their wave-lengths have
therefore no pretence to accuracy.
To visual observation these bands constitute the most
prominent and characteristic feature of the whole fringe
spectrum. It is indeed to them that the peculiar red colour
of the fringe owes its origin. In appearance these bands
are extremely hazy, no edge or other structure being-
discernible ; nor has it so far been possible to identify them
with an}* known bands, although one of them â€” namely 618,
falls very near a well-known calcium band. A higher
dispersion will no doubt solve their mystery.
262 Mr. Gr. A. Hemsalech : Excitation of Spectra of Carbon^
Luminous Vapour.
Relative intensities.
Element.
A.
Class.
Aluminium....
.. *3944-03
*3961-54
Calcium
... *3933'67
*3968-48
*4226-72
4289-36
4298-98
*4302-53
4318-64
*4425-45
*4435'32 d
*4454-78
Manganese ...
... *4030-80
403306
4034-48
Potassium ...
... *4044-15
*4047'21
Strontium ...
... *4607-34
Lead
... 3683-47
*4057'84
Chromium ...
... *4254-34
*4274"80
4289-72
Iron
... 3647-84
I.
3679-92
I.
3687-54 d
I.
3705-56
I.
3707-91
I.
3709-24
I.
*371993
I.
3722-57
I.
3727-63
I.
373332
I.
3734-86
II.
*?737-13
I.
3743-37
II.
*3745-73 d
I.
,
3748-25
I.
3749-47
11.
Luminous vapour
(graphite plate).
10
Carbon tube-
furnace (2400Â°),
Qr
12
8r
2
00
1
â€”
20
15
0 blended
" with Cr.
â€”
1
â€”
11
1
1
00
2
1
2
i
1
2
1
i
2
1
3
4
2
3
3
6
00
â€”
1
1
blended
with Ca.
000
00
00
0
00
00
2
i
0
1
o
H
00
000
1
00
0
6v*
2
3
5r
i
Ir
3
2
Titanium, and Vanadium by Tliermelectronic Currents. 263
Relative intensities.
Elemeut.
X.
Clas
Iron
3758-23
II.
3703-80
II.
3707-19
II.
3787-88
II.
3795-00
II.
3798-50
11.
3799-55
II.
3812-88
II.
3815-84
II.
*3820-44
II.
382444
I.
*3859-90
I.
3805-53
II.
3872-51
II.
3878-02
II.
3878-70 d
I.
3886-29
I.
3887-05
II.
3988-52
II.
3895-65
I.
3899-70
I.
3902-95
II.
3906-47
I.
3917-17
II.
*3920-26
I.
3922-92
I.
3927-94
I.
393030
I.
3969-26
II.
4005-26
- II.
4045-82
II.
4063-61
II.
*4071-75
II.
Luminous vapour Carbon tube-
(graphite plate), furnace (2400Â°).
1
'?
1
1
1
1
1
4
3
6 r
1
2
o
3
1
0
2
3
3
3
1
2
2
1
Since a detailed description of the spectrum emitted by
the luminous vapour is outside the scope of the present
paper, only such data have been collected in the preceding
table as were considered indispensable, both for showing
the general character of the emission and for establishing the
fact that it is of precisely the same type as that observed
in a carbon tube resistance furnace. The data regarding the
tube-furnace have been taken from an earlier communica-
tion * ; since they were obtained with the same spectrograph,
they may be directly compared with the results of the present
research : the only difference between the photographic
* Hemsalech, Phil. Mag. vol. xxxvi. pp. 218 & 224 (1918).
264 Mr. GL A. Hemsalech : Excitation of Spectra of Carbon,
records secured in the two cases is that the furnace spectra
were exposed for two minutes, whereas the spectra obtained
with the graphite plate were given only six seconds, and
therefore the relative intensities are generally on a slightly
lower scale in the latter case. As regards iron, only the
ultra-violet end of the spectrum has been given here, but
the table contains all the lines observed in that region.
Between 3824 and 3860 a few lines have not been recorded
with the graphite plate, because the strong carbon band
at 3883 rendered their observation difficult. Similarly, the
line 4216 has not been observed, as it nearly coincides with
the head of a band.
With regard to the temperature of the luminous vapour
nothing definite can be stated, as no data are available con-
cerning the temperature gradient on passing from the plate
downwards. The surface temperature of the plate was
nearly 3000Â° C. and the observations upon which the
tabular results are founded refer to the region situated
about one millimetre awajr from the radiating surface. If
we compare these results with those given by the tube-
furnace at 2400Â° C. and bear in mind the fact that the
plate results were obtained with only 1/20 the exposure of
the furnace ones, we shall probably not be far wrrong by
assuming a temperature of at least 2700Â° C. for the luminous
vapour at the distance of about one millimetre from the hot
plate. It is interesting in this connexion to mention that
under these conditions class III. group 4957 of iron was not
observed.
The great similarity between the fundamental characters
of the spectra, as given by the plate and the tube-furnace,
is so striking that there can be no room for doubt as to the
identity of the mode of excitation in the two cases. We are
therefore justified in concluding that the luminous emission
of the impurities which are expelled from the graphite and
carborundum and carried some distance downwards from
the plate, is caused by thermo-chemical excitation and is of
the same nature as that observed for these same elements
in the tube-furnace and in the mantles of various flames.
Â§ 8. Cause of the sharp outline shown by the red fringe and
of the abrupt cessation of its spectrum emission.
Sir Ernest Rutherford has very kindly directed my atten-
tion to the probable influence of the magnetic field due to
the heating current upon the motion of the electrons in the
red fringe. This magnetic field would tend to drive the elec-
trons back towards the plate and, consequently, to oppose
Titanium, and Vanadium by Thermelectronic Currents. 265
the formation of the red fringe. Sir Ernest has further
been good enollorh to calculate the strength of the magnetic
field for the particular case under consideration. Thus, with
a heating current of about 200 amperes the intensity of the
magnetic field at a point distant 0*5 mm. from the middle
of the plate is about 120 C.G.S. units, at 5 mm. from the
plate it is still 63 units? Now, with a heating current of this
strength the temperature of the plate is nearly 2700Â° C, and
under these conditions the red fringe is ordinarily not per-
ceptible to eye observations. Since, however, there is good
reason for presuming that the conductivity of the vapours in
the neighbourhood of a plate at this temperature must be
fairly high, wre can only account for the absence of the red
fringe by supposing that the thermelectronic current is being
held in check by the magnetic field set up by the heating-
current. This view seems to receive strong support by the
fact already recorded in Â§ 5 â€” namely, that the application
of a magnetic field of only about 125 C.G.S. units acting upon
the thermelectronic current in a downward sense, i e. in a
direction opposed to that of the magnetic force due to the
heating current, suffices to bring the red fringe into view.
Further, as will be shown in the next paragraph, when the
spreading downwards of the thermelectronic current is helped
by an additional magnetic field of sufficient strength, the red
fringe emission no longer stops abruptly, but dies out only
gradually. On the other hand, if in the case of the fully
developed red fringe, at a plate temperature of about
3000Â° C, an additional magnetic field be applied acting in
the same sense as that due to the heating current, the red
fringe is forced up against the plate, appearing as it were
in a state of compression, and its spectrum emission is now
seen to stop quite abruptly at the lower edge. These facts
seem to me to provide the basis for a satisfactory explanation
of the sharp outline presented by the red fringe. Supposing
there were no magnetic field set up by the heating current,
then the thermelectronic currents would be enabled to diffuse
freely through all those parts of the vapours which possess
the necessary degree of ionization. Now, since the latter is
primarily controlled by the plate temperature, the conduc-
tivity of the vapours would diminish only slowly on passing
dowmward from the plate. Consequently, the intensity of
the thermelectronic current would likewise fall off only
slowly on passing through regions of gradually decreasing
temperatures, and, as a matter of course, the red fringe
would not present a uniform band with a sharp outline, but
would fade away by degrees on passing downward from the
plate â€” a result which would be quite contrary to the observed
266 Mr. G. A. Hemsalech : Excitation of Spectra of Carbon,
facts. Thus we are led to conclude that a magnetic force
acting upwards, such as is provided by the magnetic field
due to the heating current, is responsible for the sharply
defined edge which characterizes the red fringe.
Â§ 9. Displacements of red fringe emission by transverse
magnetic fields.
In accordance with the assumption made in Â§ 7 â€” namely,
that the red fringe emission is controlled by the therm-
electronic current, its characteristic spectrum lines and
bands should be observed to lengthen or shorten whenever
a magnetic field is brought to act upon the current in a
downward or upward sense. A large number of photo-
graphic records of the fringe spectrum in the presence
of a magnetic field have been secured and all, without ex-
ception, show up the intimate connexion between the fringe
emission and the thermelectronic current. A few selected
records are reproduced on Plates III. to V. Those on
Plate III. were obtained at a temperature of about 2700Â° C.
a shows the normal development of the spectrum in the
violet and ultra-violet regions without the action of a
magnetic field, other than that due to the heating current.
As already mentioned in Â§ 5, the red fringe is not visible
under these conditions, since the red bands come out only at
the higher plate temperatures. But the carbon bands at
3883 and 4216, as also numerous titanium lines, are quite
conspicuous, although they pass down only a relatively short
distance as compared with a on Plate II., which was obtained at
a temperature of 3000Â° C. The sharply bordered continuous
emission extending downwards from the graphite plates on
a and c is due to the bluish vapour described in Â§ 4, which
in these two cases was particularly intense near the lower
edge, b shows the effect of a magnetic field of 175 C.G.s.
units acting downwards, i. e. in opposition to that set up by
the heating current. All the bands and lines of the fringe
spectrum now reach much farther down from the plate and
pass into regions where the temperature must be considerably
lower. A word of explanation is due here with regard to
the strong continuous spectrum and the marked lengthening
of the lines emitted by the luminous vapour. These are
caused, as will be shown in the next communication, by the
spiral paths along which the luminous particles move under
the combined influence of the electric and magnetic fields.
c was obtained under similar temperature conditions as b and
with a magnetic field of the same strength, but with the latter
acting upwards, i. e. in the same sense as the magnetic field
Titanium, and Vanadium by Thermelectronic Currents. 267
due to the heating current. Only traces of the carbon bands
and titanium lines are now visible in the vicinity of the plate.
The next two photographs, on Plate IV., were obtained
with a plate temperature of about 3000Â° C, at which the red
fringe is ordinarily well developed. a was taken with a
magnetic field of 590 C.G.S. units acting upwards. The rela-
tive intensification and the sharply marked boundary line of
the fringe spectrum make it appear as though the stream of
emission centres created by the thermelectronic current were
being pressed up against the graphite plate by the acting
magnetic force. This is, of course, quite in agreement with
tbe views expressed in the preceding paragraph. It is well
also to draw attention to the fact that the lines of the second
emission â€” namely, that given out by the luminous vapour
and caused by thermo-chemical excitation â€” continue to die
out gradually on passing away from the plate, in spite
of the magnetic field. b shows the fringe spectrum
under the same conditions as regards temperature and
strength of magnetic field, except that the magnetic force
is pulling the thermelectronic current downwards. With
this strong field the thermelectronic current spreads to a
very considerable distance from the plate and it is in-
teresting to note that under these conditions the lines and
bands of the fringe emission do not stop abruptly but die
out only slowly (comp. Â§8). The continuous spectrum and
the lengthening of the vapour lines are again due to the
spiral paths of luminous particles carrying electric charges.
In all the experiments described so far in this paragraph
the luminous effects due to the thermelectronic current were
observed only beneath the graphite plate and, even with a
magnetic field of 590 C.G.s. units acting upwards, no trace of
the fringe spectrum was ever seen above the plate. Stronger
magnetic fields exerted such a pull on the thin graphite plates
that the latter invariably broke before the necessary tempera-
ture was reached. But by using shorter plates, having an
effective length of only 21 mm., fields of up to 1300 C.G.s.
units could be applied for a sufficient length of time to secure
a photographic record. In this way it has been possible to
observe the red fringe above the plate and to obtain the
photograph of its spectrum which is reproduced on Plate V.
This record, I believe, constitutes the most striking and con-
vincing proof for the electrical origin of the fringe spectrum.
I was unable to ascertain whether the emission of the fringe
spectrum above the plate is due to a thermelectronic current
having passed out from the upper surface (covered with a
layer of carborundum) or, what is perhaps more likely, to
the forcing upward, round the side of the plate, of part of
268 Mr. G. A. Hemsalech: Excitation of Spectra of Carbon,
the red fringe originated below. But, in either case, the
result is of the utmost importance with regard to the cause
of the fringe emission. The strong continuous -spectrum,
due to the ed^e of the plate, is slightly widened owing to
the bending of the latter under the stress of the magnetic
force. Below the plate the spectrum shows the usual de-
velopment, except that the bands and lines of the fringe
emission are appreciably shortened by the strong magnetic
field. The lines emitted by the luminous vapour pass, as
usual, much farther down and fade away only gradually.
Now, on this side of the plate all these lines are fairly strong
as compared with those of the fringe spectrum. But in the
region above the graphite plate we find, on the contrary,
that the fringe spectrum is relatively more important than
that due to the vapours. Thus, to cite only one example,
the two aluminium lines 3944 and 3962, which, below the
plate, are much brio-liter than neighbouring titanium lines,
are, above the plate, relatively fainter than these same lines
of the fringe spectrum, although they pass to a greater
height. It is evident, therefore, that the two emissions â€”
namely, that due to thermo-electrical excitation and that
caused by thenno-chemical excitation, are differently af-
fected, and we are led to assume, either that only a small
amount of luminous vapour is present above the plate, or
that the temperature here is considerably less than what it
is below the plate. But, whichever explanation holds, this
record constitutes ah indisputable proof for the reality of
the existence of two independent emissions and, furthermore,
from the total experimental evidence furnished in the course
of this paragraph, it is manifest that the emission of the red
fringe spectrum is primarily controlled by the thermelectronic
current and not merely by the temperature of the plate.
Â§ 10. Possible cause of excitation of red fringe spectrum.
The spectroscopic evidence furnished by this research goes
to show that the thermelectronic current on passing through
the ionized vapours (probably carbides) in the vicinity of the
hot plate of graphite, is responsible only for the emission of
the spectra of carbon, titanium, vanadium, and some bands
of unknown origin. There is indeed no indication that the
lines of iron, manganese, potassium, sodium, etc., are in
the least influenced by it under these particular experi-
mental conditions, it will, however, be shown in a subse-
quent communication that when the intensity of the acting
electric field is increased (in the present case the electric
Titanium, and Vanadium hy Thermelectronic Currents. 269
field was always less than 8 Yâ€”^), the lines of other elements
become likewise affected under similar temperature con-
ditions. It will, further, be shown that the carbon bands
originate always at an anode, whereas metal lines are almost
invariably associated with a cathode. These various facts seem
to me to point to some electrolytic process as underlying the
cause of their emission, such as Professor Schuster has sug-
gested in explanation of the discharge of electricity through
gases. On this view the function of the thermelectronic
current would be to decompose molecules of carbon and of its
compounds at the surface of the incandescent plate, as well as
along the entire path marked by the red fringe and to cause
the formation of positive and negative radiating atom-ions.
Â§11. Discussion of results and their application to the case
of the electric tube resistance furnace.
The experiments described in this paper have shown most
conclusively that, when a plate of graphite is raised to a high
temperature by passing an electric current through it, the
light radiations given out by the luminous vapours in its
vicinity are due to two entirely different emissions, one of
which is controlled by the plate temperature and the other
one by the thermelectronic current. As was pointed out
in Â§ 1, such a plate may be regarded as having been formed
by cutting a tube open longitudinally and rolling it out flat.
If, now, in like manner we imagine our plate to be rolled
back, and its sides to meet and close up again, so as to form
a tube once more, would it be justifiable to conclude a priori
that the luminous phenomena exhibited by the interior of
such a tube, raised to a high temperature by passing an
electric current through it, will be fundamentally different
in character from those previously observed with the
plate ?
Furthermore, would it be possible to presume without
prejudice that all the electric actions, so manifestly promi-
nent in the vicinity of a simple plate, will, in the case of a
tube-furnace, be quite ineffective and that all the spectro-
scopic phenomena observed in such a furnace will be solely
controlled by temperature ?
In order to answer these questions we must examine
whether and in how far, the conditions which govern the
formation of the thermelectronic current, as revealed by
the experiments with a graphite plate, are fulfilled in the
270 Mr. Gr. A. Hemsalech : Excitation of Spectra of Carbon,
case of a tube-furnace. These conditions may be summarized
as follows : â€”
1. High degree of ionization of the gases and vapours in
the neighbourhood of the hot graphite;
2. An electric field of sufficient strength to drive an
electric current through the ionized vapours and
gases ;
3. A minimum magnetic field due to the heating current,
so as not to restrict the spreading out of the therm-
electronic car rents ;
and to these may perhaps be added,
4. Maximum photo-electric actions favouring the emission
of electrons.
Before discussing the particular case of the tube-furnace
it will be useful to recall the difficulties which were en-
countered in the endeavour to realize the necessary con-
ditions with the simple plate. It was shown in this case
that the loss of heat by radiation and convection was
very considerable, and therefore, in order to maintain the
plate at a high temperature, very heavy electric currents
had to be forced through it. Further, owing to the upward
rush of cool air from below the plate, the ionization of the
vapours to the desired degree was always difficult to effect,
and its accomplishment necessitated a very much higher
plate temperature than would have sufficed in the absence
of air draughts. As we have seen, the minimum plate
temperature at which the spectroscopic effects of the therm-
electronic current begin to manifest themselves is about
2700Â° C. The heating current required to attain this
temperature gives rise to a magnetic field of such magni-
tude as to seriously check the spreading out of the therm-
electronic current. Also, since photo-electric actions in the
case of a plane surface can only take place between imme-
diately adjacent points, the emission of electrons from this
cause cannot be very great. Hence we find that in the
case of an electrically heated plate, some of the es-ential
factors in the formation of the thermelectronic current are
either unfavourable or even opposed to it, and the desired
effect was only obtained by passing a heating current of high
density through the plate, entailing the establishment of an
electric field of up to 8 â€” â€” . Now, if instead of a single
L cm. Â°
plate, two parallel plates, each of only about half the thickness
Titanium, and Vanadium by Thermelectronic Currents. 271
of the former, are used, one above the other, and the
heating current divided between them, it is easy to see that
all the necessary conditions for the formation of the red
fringe will be appreciably improved. First, the better
shielding of the vapours in the space between the two
plates from the uprising air, and their exposure to a
radiating surface of twice the former area, will considerably
facilitate their ionization, which will, already at lower plate
temperatures, attain a high degree. Consequently, therm-
electronic currents will be enabled to pass under the action
of feebler electric Melds. Secondly, the heating current
through each plate will be markedly less and consequently
give rise to feebler magnetic fields ; but since the magnetic
forces due to each plate act in opposite directions in the
enclosed space, the resultant magnetic field will almost
vanish, and therefore the ionization currents will be able to
diffuse through the vapours without restraint. Lastly, any
possible photo-electric action will have a very much greater
value with a two-plate arrangement, because the two radiating
surfaces face each other. Now, as will be shown in a sub-
sequent communication, these considerations are fully borne
out by the observed facts, for with two parallel plates 5 mm.
apart the red fringe is observed already at 2200Â° C. with an
actino- electric field of only about 4 - and, further, in
Â° J cm. '
evidence of the absence of any appreciable magnetic field
within the region bounded by the two plates, the red fringe
is not confined to the immediate vicinity of the plate surface,
but fills out evenly the whole of the enclosed space. Yet,
notwithstanding the quite notable improvement observed, as
compared with the single plate, the loss of heat through
radiation and convection, in the case of two parallel plates, is
still very considerable, because the sides remain unprotected.
Hence we should expect to gain a further improvement bv
having these open sides likewise closed, and we are thus led
to anticipate that the best arrangement, fulfilling the con-
ditions enumerated, will be provided by a tube. Let us then
examine, with the help of all known facts about the tube-
furnace, in how far the conditions underlying the establish-
ment of thermelectronic currents are satisfied in the case of
a tube :â€”
1. An extremely high degree of ionization in tube-furnaces
was first shown to exist by Messrs. Harker and Kaye.
These results were entirely confirmed by my own
experiments.
272 Mr. Gr. A. Hemsalech : Excitation of Spectra of Carbon,
2. In a tube-furnace which is well protected from loss
of heat by means of a thick layer of carborundum
powder, the heating current necessary to raise the
tube to a certain temperature will naturally be less
than if the tube were not so protected ; for there are
practically no heat losses to make good as compared
with the single plate or even the two-plate arrange-
ment. Consequently, with such a tube-furnace, the
drop of potential along the tube will be relatively
small and, in my particular case, it was only about
1*05 T L for a temperature of 2600Â° C, as against
cm. r ' to
5*5^â€” s in the case of the sing-le plate. Hence, the
cm. Â© i >
acting electric field in a tube-furnace protected
against heat losses in the manner stated is rela-
tively small and may even not suffice, especially at
the lower temperatures, to produce all those spectro-
scopic effects which were observed with the single
plate.
3. The magnetic field inside a tubular conductor is zero *.
Thus, if the conductivity of the vapour has the
necessary value, thermelectronic currents will pass
absolutely unimpeded by magnetic forces and be con-
trolled solely by the degree of ionization prevailing
in the interior.
4. Photo-electric actions should be very great in this case
since every point is acted upon from all round the
circular wall.
Thus we see that, with the possible exception of con-
dition 2, there is a great likelihood that the electric actions,
which have been shown to underlie the luminous phenomena
constituting the red fringe, should also prevail in the interior
of a tube-furnace.
Let us then examine more closely in how far the spectro-
scopic results obtained with the tube resistance furnace
agree with my observations on the spectrum of the red
fringe. Since the magnetic field inside the tube-furnace is
zero the red fringe, or its equivalent, should fill out the whole
space, as in the case of the two parallel plates. Tt will be
remembered that I found the light given out by the interior
of the tube-furnace, up to a temperature of irom 2400 to
'2500Â° 0., to be of purple tint f- This., of course, suggested
* Sir J. J. Thomson, Elements Mathem. Tlieoiy of El. & Masrn.,
3rd ed. p. 333.
f Hemsalech, Phil. Mag. vol. xxxvi. p. 214 (1918).
Titanium, and Vanadium by Thermelectromc Currents. 273
the emission of red radiations. Now, a re-examination of
my old spectrograms referring to the tube-furnace has
revealed the presence of the characteristic bands of the red
fringe; but although they coincide in position, their character
is markedly different from that observed with the plate, for
they are much less intense and, what is more noteworthy,
nothing like as hazy. To give an idea of their relative
unimportance in the case of the tube-furnace, it may suffice
to mention that the more sharply defined calcium bands at
6185 and 6211, which are completely masked by them in the
spectrum of the red fringe as obtained with the single plate,
are distinctly brought out. It will be shown in a subsequent
paper that, when two parallel plates are used, the red bands
of the fringe spectrum, although much more intense and
prominent than in the tube-furnace, are nevertheless appre-
ciably reduced in haziness and extent as compared with
their development in the case of the single plate. Now, the
only vital factor which changes appreciably on passing from
the single plate to the two-plate arrangement and the tube-
furnace, is the intensity of the electric field. Thus, with
the single plate, the red bands begin to show only with an
electric field of about 7 â€” -; with the two-plate arrangement
cm. * a
they come out with only about 4 -^-, and in the tube-furnace
already with less than 1 â€” ; the diminishing values for the
intensities of the electric fields are, of course, in keeping
with the higher degrees of ionization which prevail in the
case of the two parallel plates and in that of the tube-furnace.
These facts and considerations seem to indicate that the red
bands, being as they are stimulated by electric actions, will
be susceptible also to variations of the acting electric force,
and it may, therefore, reasonably be conceived that the
extraordinary haziness and intensity which characterize
them in the red fringe are caused by the relatively strong-
electric field which is created by forcing a heavy current
through the graphite plate. I hope to show in a subsequent
paper that the lines of the doublet series of sodium are simi-
larly affected when acted upon by strong electric fields.
Hence the weakness and the relatively low degree of hazi-
ness of the red bands in the tube-furnace are quite consistent
with the feeble electric field prevailing therein.
The lines of titanium and vanadium have not been ob-
served by me in the tube-furnace ; it is possible that these
elements are not contained as impurities in the carbon tubes
Phil. Mag. S. 6. Yol. 39. No. 231. March 1920. T
274 Mr. Gr. A. Hem sale ch : Excitation of Spectra of Carbon,
employed. But it is of interest to note that the lines of
these elements have been recorded as impurities by Dr. King
in a tube-furnace made of the same material as my plates â€”
namely, Acheson graphite *. This important observation
poiuts most forcibly to the presence of thermelectronic
currents in a tube resistance furnace.
We now pass on to examine the relative behaviour of the
remaining constituents of the red fringe spectrumâ€” namely,
the carbon bands. These, as was shown in Â§ 7, consist of the
so-called cyanogen bands and those of the Swan spectrum.
The strong band at 3883 is the first to appear as the tem-
perature of the plate gradually rises, and its first traces
were detected already at 2500Â° C, whereas the complete
spectrum was observed only at about 3000Â° C, the Swan
bands being the last to come out. In conformity with the
more effective ionization of the vapours, and the absence
of an internal magnetic field in the tube-furnace, we shall
naturally expect to observe the carbon bands in the latter
already at much lower temperatures than with the single
plate. Now, according to my own observations f , the first
trace of the carbon spectrum â€” namely, the head of the
so-called cyanogen band at 3883, was noted already at
a temperature of only about 1900Â° O. As the furnace
temperature and consequently also the acting electric field
were increased, the carbon spectrum developed further by
the additional emission of the band at 4216. Finally, at
temperatures of above 2500Â° C, the bands of the Swan
spectrum came also into view and thus the spectrum attained
a similar degree of completeness to that observed in the red
fringe. From these observations we elicit the double fact
that the carbon spectrum emitted by the tube-furnace is not
only of precisely the same character as that given b}^ the
red fringe, but also that its gradual development with
increase of temperature and electric field proceeds in exactly
the same order as that noted therein. As I have already
pointed out, this order of development is just the reverse to
that observed when the emission is caused by chemical
excitation, such as prevails in the explosion regions of the
various hydrocarbon flames. Hence the emission of the
carbon bands in the tube-furnace cannot be ascribed to
chemical actions of this nature. Further, it cannot possibly
be due to thermal or thermo-chemical actions alone, because
none of these bands are emitted by the hottest regions in the
mantles of carbon flames. Finally, it has been conclusively
* A. S. King-, Astrophysieal Journal, vol. xxxvii. p. 250 (1913).
t Hemsalech, Phil. Mag. vol. xxxvi. p. 225 (1918).
Titanium, and Vanadium by Thermelectronic Currents. 275
proved that, in the red fringe, the emission of the carbon
bands is primarily governed by electric actions, the tempera-
ture serving only to ionize the vapours in the vicinity of the
plate. Hence, since both the character and the order of
development of the carbon spectrum as given by a tube-
furnace are fundamentally identical with those observed in
the red fringe, we can only conclude that their emission by
the former is due to the same cause â€” namely, the passage
of thermelectronic currents through highly ionized vapours
and gases. Thus, if it were possible to exclude therm-
electronic currents from the interior of the tube, no carbon
bands should be observed, as is the case in the mantles of
hydrocarbon names up to 2700Â° C.
The arguments brought forward in this discussion lead to
the inevitable conclusion that thermelectronic currents, just
as in the case of the red fringe, are responsible also for the
-emission, in the tube-furnace, of the carbon bands and
the red bands of unknown origin. The mere fact that
a plate has been rolled up into a tube can therefore not in
the least affect the fundamental character of the luminous
phenomena observed in its vicinity ; on the contrary, several
of the essential conditions for their production are thereby
greatly improved. The relatively feeble development of the
red bands in the tube-furnace can be most satisfactorily
ticcounted for by the feebler electric field existing in the
latter ; and this view is entirely corroborated by rny observa-
tions with a parallel-plate furnace, in which both the
intensity of the acting electric field and the state of develop-
ment of the red bands were intermediate between those
noted with the single plate and the tube-furnace.
With regard to the spectrum emission of the luminous
vapour, the lines and bands of which are observed to extend
below the red fringe spectrum, it has been proved in Â§ 7
that it is of the same character as that emitted by these
vapours in the tube resistance furnace and in the mantles of
flames under the influence of thermo-chemical excitation.
The present experiments with the single plate have, there-
fore, also furnished an independent confirmation of the
-existence of this emission in the tube-furnace.
Â§ 12. Probable cause of disagreement between Dr. King's
results and mine.
I have shown * that a tube resistance furnace, worked
with continuous current, emits cla>s III. lines of iron only at
* Hemsalecli, Phil. Mas', vol. xxxvi. p. 214 (1918).
T2
276 Mr. Gr. A. Hemsalech : Excitation of Spectra of Carbon y
temperatures above 2500Â° C; even of the bright group at
4957, no trace was ever detected below 2500Â° C. I have
further pointed out, that this observation is quite in harmony
with the results of my investigations on the flame spectra of
iron. In this case also, there was no sign of class III.
group 4957 up to the temperature of the oxy-hydrogen
flame â€” namely, 2550Â° C, and only mere traces of it were
observed in the oxy-acetylene flame at 2700Â° C. But, when
the temperature of the tube-furnace was raised to 2700Â° (J.,.
the spectrum of iron emitted by it, besides being of great
brilliancy, showed a remarkably high degree of development,
with group 4957 as quite a conspicuous feature ; in fact the
character of the spectrum had completely changed and now
approached more that stage which it generally attains in
the arc. Thus, the spectrum of iron, as emitted by my tube-
furnace at 2700Â° C, was found to be entirely different from
that given by the oxy-acetylene flame at the same tem-
perature. Had the emissions of the iron spectra in these two
cases been solely controlled by temperature, then surely
the spectra given by these two sources would have been
identical, at least in so far as the relative intensities of the
lines were concerned. From the very fact that this was not
the case, and also from several other considerations, I had
arrived at the conclusion that the emission of class III. lines
by the tube-furnace was caused by part of the heating
current passing through highly ionized iron vapour, and
that the iron spectrum observed under these conditions
should be regarded as a low-tension arc spectrum. This
conclusion appeared to receive confirmation by the negative
results of an experiment with a special plate-furnace, in
which the acting electric field was greatly reduced. No
trace of class III. group 4957 was ever seen with this
furnace, although the observations were continued up to the
moment when the plate burnt through and the temperature,
as is well known, had reached a maximum value. As
Dr. King has rightly pointed out, I ought to have observed
at least those lines which are caused by the action of heat
upon the carbides of iron ; but unfortunately I had been
unable to obtain a photographic record of the spectrum on
account of a breakdown of the dynamos, and I have no
doubt that such a record would have shown class I. and II.
lines. As it was, however, I had concentrated my attention
upon the detection of group 4957 of iron and the Swan
bands of carbon, neither of which were observed. The
strong continuous spectrum, ahvays present, may possibly
have contributed to render the other lines less conspicuous,
'Titanium, and Vanadium by Tliermelectronic Currents. 217
but there was also as intense a continuous spectrum emitted
by the high-temperature tube-furnace, yet, in spite oÂ£ it. the
complete iron spectrum, including class III. lines, stood out
most brilliantly.
Now, my observations do not apparently harmonize with
the results obtained by Dr. King, who has photographed
group 4957 of iron already at temperatures of between
1800Â° and 1900Â° C, and in one case even at 1650Â° C.*
There can be no doubt that the range of temperatures given
by Dr. King's furnace is the same as that available with
mine â€” namely, from about 1500Â° to 2700Â° C, and, if the
emission of class III. lines of iron were solely controlled by
the temperature comprised within these limits, I should most
certainly have seen them, or secured them photographically
on one or other of the many records taken. The double
fact, that class III. lines of iron were not observed in my
furnace, nor in the mantles of flames at corresponding*
temperatures up to about 2500Â° C, seems to me to indicate
clearly that there must be some other factor present in
Dr. King's furnace which is responsible for the emission
of these lines at low temperatures. Let us then examine
more closely whether there is any fundamental difference
between the furnace arrangement used by Dr. King and
mine. As will be remembered, my furnace was formed by
a carbon tube of 14 mm. internal diameter and an effective
length of about 10 cm. The heated portion of the tube was
well protected against loss of heat by means of a thick layer
of carborundum powder. The heating current supplied to
the tube was continuous. The drop of potential along the
tube, for temperatures of from 2600Â° to 2700Â° C, was about
1'05 â€” . On the other hand, Dr. King's furnace consists
cm. ' fe
of a tube of Acheson graphite having 12*5 mm. internal
â€¢diameter and an effective length of about 20 cm.f The tube
is not protected by a carborundum jacket, but is enclosed in
-a steel chamber, which is pumped out to a pressure of less
than 2 cm. of mercury. Owing to the greater loss of heat
through radiation entailed by this arrangement, it requires
a larger current to produce the same furnace temperature
as with a well protected tube, such as is the case with my
furnace. Accordingly, a potential of 30 volts has to be
applied to Dr. King's tube in order to bring the temperature
to between 2600Â° and 2700Â° 0. This corresponds to a
* A. S. King, Astroph. Journ. vol. xlix. p. 48 (1919).
t A. S. King, Astroph. Journ. vol. xxxvii. p. 120: ibid. p. 240 (1913).
278 Mr, G. A. Hemsalech : Excitation of Spectra of Carbon,
potential gradient along the tube of about 1*5 ^. Further-
more, Dr. King's furnace, unlike mine, is worked with
alternating current, and the instruments employed in
measuring volts and amperes give, of course, only the root-
mean-square values of these quantities. Now, the value for
the potential gradient along the tube reaches two maxima
during each complete cycle of an alternation, which are
equal to 1*4 times the root-mean-square value as read on the
voltmeter. In the case of Dr. Kino's furnace this amounts
volts
to 1*4 x 1*5 = 2*1 â€” . for a furnace temperature of between
2600Â° and 2700Â° C. In like manner, at a temperature of
between 1800Â° and 1900Â° C, a potential of 15 volts has to be
applied in order to drive the necessary heating current
through the tube, and the maximum value of the potential
gradient works out to 1*05 â„¢\ For the sake of con-
venience the numbers found for the potential gradients
along the furnace tubes in the two cases are restated in
the following table
Continuous current
tube well protected
Alternating current
with carborundum tube
powder. unprotected.
Temperature. (Hemsalech.) (Dr. King.)
1800-1900Â° C. Not recorded. 1-05â€” 8.
cm.
2600-2700 â€ž 1-05T~ 2-1 â€ž
These figures reveal the fundamental difference between
Dr. King's and my furnace â€” namely, the acting electric
field is twice as powerful in the unprotected tube run with
alternating current as in the protected tube worked with
continuous current. Already at the lower temperature
of between 1800Â° and 1900Â° C, Dr. King's furnace gives an
electric field of such strength as mine attains only at between
2600Â° and 2700Â° 0. Hence it is quite evident that for a
given furnace temperature thermelectronic currents will
reach a much higher development in Dr. King's furnace
than in mine, and we must, therefore, expect that his
furnace will show spectrum lines and bands which cannot
be seen in mine under the same temperature conditions.
Thus Dr. King's furnace, thanks to its stronger electric
field, will show lines at the low temperature which my
furnace, owing to its feebler electric field, only shows at the
high temperature. This is precisely what has been observed
Titanium , and Vanadium by Thermeleetronic Currents. 279
with regard to class III. group 4957 of iron. The stronger
electric actions in Dr. Kin^s furnace have brought this
group out at already 1800Â° C, whereas in my furnace it
shows onty at over 2500Â° C. when the acting electric field
reaches about the same value as that prevailing in Dr. King's
furnace already at the lower temperature. Further, the
enhanced spectroscopic effect due to the stronger electric
actions in the alternating current furnace, is proved beyond
doubt by Dr. King's observation that in his furnace the
cyanogen bands are strong at 1850Â° C* Now, these bands,
which, as has been firmly established in the course of the
present research, are directly excited by the thermelectronic
current, are feeble in my furnace at about the same tempera-
ture, in fact only the head of the brighter band â€” namely
3883, has been detected at about 1900Â° C.
It might be contended witli regard to the electric effect
of the alternating heating current, that the short duration of
the maximum value of the voltage reached twice during
each cycle may not suffice to appreciably affect the excitation
of spectrum radiations. Against this we have, however, the
very important experimental evidence by Dr. de Watteville
who, by a most ingenious method, has shown that the
spectrum emission follows very closely the periodic varia-
tions in an alternating current circuit f. But in this
connexion it is well to draw attention to the possibility of
the furnace temperature being likewise subject to periodic
fluctuations, owing to the rise and fall of the intensity of the
heating current. The temperature, as indicated by a pyro-
meter, would represent the average during a cycle ; but it is
just possible that the temperature, and therefore also the
degree of ionization, may momentarily reach a much higher
value than this, and this higher value would occur nearly
simultaneously with the maximum of the acting electric field.
Thus, in certain circumstances the running of an electric
furnace by means of alternating current may, also from this
point of view, affect the estimation of the true character of
the spectra emitted by the luminous vapours. It is probable,
however, that in a well protected furnace tube, temperature
variations of the kind here suggested will not be of great
amplitude; but, in the case of an unprotected tube, which
loses heat rapidly through radiation, such as is precisely the
cise in Dr. King's arrangement, the possible effects upon
temperature and ionization of the periodic variations in the
* A. S. King-, Astroph. .Tourn. vol. xlix. p. 50 (1919).
t C. de Watteville. Comptes Rendu s de I' 'Academie des Sciences, seance
du 22 ievrier, 1904.
280 Mr. Gr. A. Hemsalech : Excitation of Spectra of Carbon,
intensity oÂ£ the heating current, should receive careful
attention.
When considering the action of an alternating current
furnace it is necessary also to bear in mind, that the diffusion
of the thermelectronic current through the ionized vapours
in the interior might, to some extent, be influenced by the
skin effect. This would, under certain conditions, cause
the complete stoppage of electric currents near the axis
of the tube. But it is to be presumed that, with ,the low
periods generally employed for heavy alternating currents â€”
namely from 50 to 100 per second, the thermelectronic
currents will nearly reach the centre. Dr. King has made
some interesting experiments which directly bear on this
point *. Two graphite exploring rods were inserted into the
tube from opposite ends. The inner end of one rod was
placed in the hottest part of the tube, and that of the other
rod in the cooler region near the end. A direct current
ammeter, placed in the circuit connecting the two exploring
rods, registered a current of about 1*5 amperes at a furnace
temperature of about 2600Â° C. The rod in the hot part had,
of course, become a cathode and, together with the cooler
anode, acted as a rectifier for the alternating current, part of
which leaked across the space from the inner wall to the
rod in the centre. Thus there *is little doubt that in
Dr. King's furnace thermelectronic currents will be enabled
to pass very near to the axis of the tube. Dr. King has
further shown that the degree of ionization decreases rapidly
with increasing pressure ; it is therefore important to record
the fact that most of Dr. King's observations on furnace
spectra were made at low pressure, whereas all mine have
been carried out at atmospheric pressure.
With regard to the line emission of iron which is caused
by thermo-chemical excitation, there is little doubt that its
state of development at a given temperature, in the alter-
nating current furnace, will be the same as that observed in
the continuous current one. But, with the former type of
furnace, especially when the tube is unprotected, the second
line emission â€” namely, that due to thermo-electrical exci-
tation, becomes perceptible already at the lower furnace
temperatures ; consequently, the iron spectrum which is
observed under these conditions will, already from the
outset, be the result of the superposition of two different
* A. S. King, Astroph. Journ. vol. xxxviii. p. 831 (1913).
Titanium, and Vanadium by Thermelectronic Currents. 281
emissions. Now, since both the furnace temperature and
the intensity of the acting electric field will rise simul-
taneously when the heating current is increased, the
progressive developments of the two emissions will likewise
proceed simultaneously, and it is no doubt owing to this
fact that the composite character of the iron spectrum, as
given by an insufficiently protected alternating current
furnace, has escaped the attention of Dr. King. For, as
I have shown, with a well-protected continuous current
furnace, though the range of temperatures produced is the
same, the second emission, on account of the much feebler
electric field, manifests itself only at temperatures of over
5500Â° C. â€” namely, after the first emission has already
attained a high degree of development. And it is probably
thanks to the smaller electric field of my furnace that such
good agreement was found to exist between the furnace and
flame spectra of iron up to about 2500Â° C, because the
spectrum of this element as emitted by my furnace up to
this temperature was entirely caused by the action of heat
on a compound of iron under the sole control of the prevailing
temperature.
Â§ 13. Summary of Results.
1. The existence has been established of a new luminous
phenomenon which manifests itself as a sharply defined
band, of pinkish hue, in the immediate vicinity of a
plate of graphite raised to a high temperature by
means of an electric current. The name red fringe
has been applied to this luminous band. Â§ 4.
2. It is shown that the red fringe is caused by an electric
current, which is probably part of the heating current.
3. The red fringe is not solely controlled by the plate
temperature, for it can be made to appear or disappear
at will by the simple action of magnetic forces, without
altering the temperature. Â§ 5.
4. It is shown that the electric current underlying the
formation of the red fringe is quite distinct from
thermionic currents, not only as regards sensitiveness
to the magnetic force, but also and particularly as to
the spectroscopic effects produced. In order to clearly
distinguish between the thermionic current and that
which gives rise to the red fringe, it is proposed to call
the latter the thermelectronic current. Â§ 6.
2S2 Mr. G. A. Hemsalech : Excitation of Spectra of Carbon,
5. Spectroscopic examination has revealed the existence of
two entirely different and distinct emissions by the
luminous vapours in the neighbourhood of the heated
plate. One of these emissions is controlled by the
plate temperature and is due to thermo-chemical
excitation. The lines and bands which compose its
spectrum are observed to die out only gradually on
passing away from the plate. The second emission is
confined to the immediate vicinity of the plate, and
in extent it coincides with the red fringe. It is
further shown that this emission is controlled by the
thermelectronic current and is therefore due to thermo-
electrical excitation; The characteristic feature of its
spectrum is that its lines and bands, unlike those of
the first emission, stop quite abruptly at the lower
edge of the red fringe. Â§ 7 and Â§ 9.
6. The spectrum of the first emission (thermo-chemical
excitation) is composed of the lines or bands of the
more volatile metals, such as Na, Li, Sr, K, Al, Oa,
Mn, Fe, etc. ; that of the second emission (thermo-
electrical excitation) is constituted of the bands of
carbon, the lines of titanium and vanadium, and a
group of characteristic red bands of very hazy
appearance. Â§ 7.
7. The gradual development of the carbon spectrum with
thermo-electrical excitation is in the reverse order to
that observed with chemical excitation. It is further
pointed out that this spectrum is not emitted by the
hottest parts of the mantles of the air-coal gas, oxy-
coal gas, and oxy-acetylene flames. Consequently, it
is not excited by thermal or thermo-chemical actions
up to a temperature of 2700Â° 0. Â§ 7.
8. The sharp outline presented by the red fringe is caused
by the magnetic field around the plate due to the
heating current. This magnetic field has a tendency
to drive the thermelectronic current back towards the
plate, and it thus prevents the diffusing downwards of
even the least part of this current. Â§ 8 and Â§ 9.
9. A discussion of the thermal, electric, and magnetic con-
ditions existing in the vicinity of an electrically heated
plate, as compared with those prevailing in the interior
of a tube resistance furnace, has led to the inevitable
conclusion that thermelectronic currents will be enabled
to pass through the ionized vapours in the latter already
at a comparatively low temperature, and moreover,
they will diffuse without difficulty throughout the
Titanium, and Vanadium hy lliermelectronic Currents. 283
whole inner space of the tube, because the magnetic
field due to the heating current is zero. Furthermore,
the character and order of development of the carbon
spectrum in the tube-furnace, as compared with that
in the red fringe, indicate plainly that, also in the
furnace, its emission is caused by thermelectronic
currents. Â§ 11.
10. It is pointed out that the disagreement between the
furnace results obtained by Dr. King and myself is
most probably due to the different arrangements em-
ployed. Dr. King's furnace tube, being unprotected,
requires a greater heating current and consequently
a stronger electric field to raise it to a given tempera-
ture than is necessary for a furnace tube which is well
protected against heat losses by means of a carborundum
jacket. Further, by using alternating current for heating
the tube, the electric field reaches a maximum value
of 1*4 times that given at the same temperature with
continuous current heating. For these two reasons
the maximum acting- electric field in Dr. King's
furnace is twice as strong as in mine for any given
temperature. Hence, lines or bands the origin of
which can be directly traced to the action of therm-
electronic currents, such as the carbon bands and
class III. lines of iron, will in the unprotected alter-
nating current furnace show already at much lower
temperatures than in a well-protected continuous
current furnace. Â§ 12.
Â§ 14. Concluding Remarks.
The present investigation has furnished most conclusive
results in evidence of the emission, by the tube resistance
furnace, of luminous radiations which depend for their exci-
tation upon electric actions. It has further been shown that
the spectroscopic effects produced in this way are governed
by the acting electric field and the state of ionization of the
vapours. They depend upon the furnace temperature onlyr
in so far as the latter controls the electrical conductivity of
the vapours. If the furnace be well protected from loss of
heat and be worked with continuous current, the electric
field which is established at the lower temperatures may be
too small to stimulate those atomic vibrations which are
sensitive to electric actions. This has been shown to be the
case with iron. If, on the other hand, the tube of the fur-
nace be left unprotected and be heated with alternating
284 Spectra of Carbon, Titanium, and Vanadium.
current, the electric field will, already at the lower tempera-
tures, be of sufficient strength to give rise to traces of the
electrically controlled emission. Now, since this emission
naturally develops progressively, as the temperature and
consequently also the electric field increase, the erroneous
impression will be created that the observed spectrum (which
is of course composed of the electrically controlled emission
and that due to thermo-chemical excitation) is entirely
governed by the furnace temperature.
It seems to me therefore highly probable that the electric
tube resistance furnace, and in particular the special furnace
arrangement used by Dr. King, is most unsuited to investi-
gations having for object the study of the effect of tempera-
ture upon the spectrum lines of an element, because it is not
possible with such furnaces to discriminate readily between
the lines controlled by temperature and those, the emission
of which depends upon electric actions* Hence, spectrum
tables which are founded solely on furnace observations and
purpose to give the variation of the spectrum with tempera-
ture, are necessarily misleading, since in their compilation
no account has been taken of the composite character of the
spectrum observed. One of the objects of experimental
spectroscopy is to break up a complicated spectrum and to
trace the origin of the various component parts, in order
to pave the way for the search of still closer relationships
between the various lines or groups of lines. This can often
only be accomplished by lengthy comparative investigations
under most diverse conditions. In this way I have shown,
that the particular furnace emission of iron vapour which is
controlled by temperature, is obtained by itself in the mantles
of flames, where it can be observed to much better advantage
than in the tube-furnace.
Hence it seems to me that the study of furnace spectra
can only lead to trustworthy and useful results, if it be sup-
plemented by an examination of both arc and flame spectra
of the same element, the whole investigation being of course
made, as far as possible, with the same spectrographs
applicances.
Manchester, July 1919.
Note added November 21.
The hazy red bands of the fringe are due to the oxide and
more especially the carbide of calcium. The oxides of
strontium and barium when added to the carborundum, with
fundamental Equation in general Theory of Relativity. 285
which the graphite plate is covered, give bands of similar
appearance and likewise sensitive to thermo-electrical ex-
citation. The strontium bands are situated in the red and the
blue, and those of barium in the green. With the oxide oÂ£
the last named metal the colour of the fringe is green. If
none of these compounds is present in appreciable quantities
the fringe has a bluish-violet tint.
The spiral and helical paths observed when a magnetic
field is acting downwards are caused by the motions of
positively charged particles emitted by the alkali metals.
The temperature at which emission of positive particles
begins varies inversely as the atomic weights of these
elements. The deflexion of the positive streamer by a
magnetic field provides a very sure guide as to the presence
of these elements in minerals, &c; and this method of
detection is more rapid and convenient than the spectro-'
scopic test. There are also indications of the emission of
negative particles at higher temperatures and with stronger
magnetic fields. It seems likely that the phenomena ob-
served are analogous to those discovered by Sir J. J.
Thomson in connexion with the emission of rays of positive
electricity by gases at low pressures*. Thus the emission
of positive particles by the alkali metals might possibly be
caused by the breaking up of the molecular or atomic systems
of these elements, which according to preliminary deter-
minations would set in at approximately the following tem-
peratures: lithium at 2700Â° C. ; sodium, 2550Â°; potassium,
2300Â°; rubidium, 2200Â° ; and caesium, 1900Â°.
XXYI. Parametric Solutions for a Fundamental Equation in
the general Theory of Relativity, with a Note on similar
Equations in Dynamics. By E. T. Bell f .
1. TT may be of interest to note that we can solve
A ds2 = talJdxid.rj (i,j=l, 2,3,4), . . . (1)
which is the point of departure for much recent work in
the theories of generalized relativity and gravitation, very
simply if we assume that s, x\, a'2, x3i #4 are functions of four
* J. J. Thomson, Phil. Mag. vol. xxi. p. 22o (1911).
t Communicated, by the Author.
286 ProE. E. T. Bell on Parametric Solutions for a
arbitrary functions
&=Â«Â«) 0 = 1,2,3,4), .... (2)
where u is a parameter. In (1) the a{j are functions oÂ£
the Xi.
2. For brevity we shall assume that the right-hand
member of (1) is reducible, in the usual way, to a sum of
four squares,
4
(U2 = Â£ ( hn dxl + bi2 ch\2 + bi3 dxs + bi4: dx^f, . (3)
subject to the condition that the determinant of the system
budxl + bi2dx2 + bisdx, + b{i dxA = ,- (u) (i = 1, 2, 3, 4), . (I)
in which dx^ dx2i dx%, dxÂ± are the unknowns, does not
vanish. If (1) is such that this reduction is impossible ;
viz., if the aij are such that no by exist such that the deter-
minant of (4) is different from zero, a slight and obvious
modification of the algebra leads to a parametric solution of
essentially the same sort as that now given.
3. The determinant of (4) not vanishing, we can solve for
the dxi, getting
dZi^Bnfafa) +Btofa(u) + B&z(u) +B^(w) (t = l, 2, 3, 4),
... (5)
in which By are functions of the &#, hence of the a^ and
therefore of the Xi alone. Hence, integrating,
^â€¢^(Bfl^^ + Ba^^ + Bis^sW + Btt^WyM ; â€¢ (6)
or when the J$i-3 are independent of u,
4
^SB^(Â«)iu (i=l, 2,3,4). . . (7)
We shall now determine 5 and the /(V) in terms of the
functions (2) so that on substituting in (6), 5 and the
resulting xi reduce (1) to an identity. There are many
ways of doing this ; one of the most obvious is
*=J(?i2+&2+ ft2 +?/)*',
Fundamental Equation in general 1 lieory oj Relativity . 287
For, on putting these values of the (j>//u) in (6), and then
going back successively through (6), (5), (3), we get for the
right-hand member of (3),
(?i2 + it - U + HY- + (Sf.fs)2 + (2f2fs)2 + (2f4f3)2,
which is identically
that is, ds2. Hence (3), and therefore (1) which is equiva-
lent to (3), is identically satisfied by the indicated values of
â– s, %U #2> #3> Xi'
4. The same device of making the solution depend ulti-
mately upon an identity connecting sums of squares, can
obviously be applied to find parametric solutions of other
equations occurring in general dynamics. It will be suffi-
cient to indicate the identity applicable to
ds2 = ^aijdXidxj (i,j=l, 2, . . ., re). . â– . (8)
Suppose that the quadratic form on the right of (8) is
algebraically reduced, in the ordinary way, to a sum of
n squares,
n
ds2 = ^(ljiid,vl + bi2d.v.2+ . . . +biHdlvny, . . (9)
such that the determinant |6^| does not vanish. As before,
special cases in which this reduction is impossible may arise ;
but they present no essential difficulty. Then, for n>l, we
resolve reâ€” 1 in any way into a pair of factors r, s so that
?i = rs+l ; and put
Ar=,2Â£23 B.=ii&Â»,
where f;, rjj are arbitrary functions of a parameter u. Then
the identity leading to a solution of (9), and hence of (8), is
(Ar + BS)2 = (A,.-B5)2 + 4A,B5. . . . (10)
For, on multiplying out A?BS, the right of (10) is a sum of
rs + l = ?i squares, viz.,
(Ar + B8)2 = (Ar-Bs)2 + k Z(2%iVjy;. . (11)
i=l ;'=1
288 Fundamental Equation in general Theory of Relativity*
and hence on putting
s=Â§(Ar + Bs)du,
hi^ + bi2x2 + . * . + bin Â®n =J ; (w) du (i = 1, 2, . . ., n),
. . . (12)
where the functions
V cy 1
c
For the first harmonic -pâ€” â€” ^. By the use of a calibrated
variable condenser Co any suitable smaller relation j^j-r
can be chosen. Thus the number of maxima to be counted
is reduced and the operation with the circuit IV spared.
The application of this method to closed circuits con-
taining known capacities and inductances gave results
in good agreement with those calculated by Thomson's
formulaa.
The above method was also used for the exact deter-
mination of wave-lengths emitted by vertical antennae
and other radiating circuits, with distributed capacity and
inductance, used in radio-telegraphy.
On board the ' Lorraine/
Aug. 30, 1919.
XXVIII. Variably Coupled Vibrations: Gravity-Elastic Com-
binations. Masses and Periods equal. By L. 0. Jackson,
F.P.S.L., University College , Nottingham *.
[Plates VI. & VII.]
I. Introduction.
I^HE present paper is a continuation of the work of
Prof. Barton and Miss Browning (Phil. Mag. vol.xxxiv.
p. 246, vol. xxxv. p. 62, vol. xxxvi. p. 36) on the subject of
coupled vibrations.
The coupled system treated in the following pages consists
of an elastic lath pendulum and a gravity pendulum, which
can be attached to the lath at different points along its length.
The degree of coupling of the pendulums thus depends
* Commrmicated by Prof. E. IT. Barton, F.R.S.
Variably Coupled Vibrations.
295
on the position of the point of suspension of the gravity
pendulum*.
The paper includes 32 photographic traces of the motion
of the gravity pendulum under various conditions of coupling
and starting.
II. Theory.
Equations of Motion and Coupling,
For the gravity and elastic pendulums, let the masses of
the bobs be M and N, let the length of the simple pendulum
equivalent to PM be r, and at time t let the linear displace-
ments of M and N be y and z respectively ; further, let the
Fig. 1.
linear displacement of P be olz. It should be noted that
the displacement of the bob N oE the elastic pendulum has
always the positive sense as shown in fig. 1.
Then for small oscillations, and considering PN as straight,
the equations of motion may be written
"2*
*(*==) -o,.
S5J+.HA = 1^Â»=5).
* The bob of this gravity pendulum carried an electric lamp and
a lens, and left on a plate below it a photographic record of its motion.
296 Mr. L. C. Jackson on
These may be re-written :
M^ + Mm2?/ = Mm*uz, . . . (1)
^ji+ (Nn2 -4- M?nV> = Mm*Â«y, . . . (2)
where ??i and n are derived from the free isolated vibrations
of M and N respectively, viz. :
y = a sin mt and z = b sin nt,
and 9 #
m2 = ~. t
Following the analogy of electrical practice, we may write
the coefficient of coupling 7 as given by
2 _ _MmV_
'y-N^ + MmV * ' '3)
Solution and Frequencies.
To solve (1) and (2;, try in (1)
y = <* â€¢ â€¢ â€¢ â€¢ iÂ£)
This gives ; = ^ + ^^ (5)
Then (4) and (5) in (2) give the auxiliary equation in x :
NV + x2(l$m2 + N;i2 + Mm2*2) + Nn2m2 = 0. . . (6)
This may be re-written in the form
xÂ± + w2(p2 + q2)+p2q2 = 0. .... (7)
Hence x = Â±pi or +g-i. . . .' . . . (8)
From this point we shall treat the case in which M = N,
m = n.
Thus the equation (3) for the coupling now reduces to
2_ *2
7 ~"l-fÂ«2*
Hence, on inserting the usual constants, we may write for
the general solution and its first derivatives,
y = E sin (pt + e) + F sin (qt + ), (9)
9 5 9 2
r â€” ra* . . mz â€” c
â€” 5 â€” iii sin (pt + e) + 5-=
T^-Brin (?i + e) + V-J sin (?Â« + <#Â»), (10)
Variably Coupled Vibrations. 297
iv
j = pK cos (j>t + e)+qF cos (^ + <Â£), (11)
dt m'2ot J *~ " v/ " 7 7?i2a
rj~ lit lit Of
- -2 â€” jÂ»E cos (/>Â£ + e) + j-4- A- {2m2 + m2cc2) p2 -f m4 = 0. . . (14)
Thus, calling the larger root pr and the smaller q2, we
have
_ ,Â»'t(2 + ,') + [(2 + Â«')'-4]'}
V â€” .) , . . . [lO)
whence
ga_,n'{(2 + qÂ»)-[(2 + a')'-4]*K _ _ _ (16)
P=i(2 + *')+[(2 + cSf-i]il 17)
9 {(2+^-[(2 + Â«^-4]*}** â– 'â– .'."
Initial Conditions.
(i.) Lower bob struck.
AVe ma)r here write
y = 0, z = 0, || = Â«, J = 0 for t = 0. . (18)
These inserted in (9) to (12) give equations satisfied by
, = 0, = 0, E=^I^, F = (^-^. (19)
(p2-g2)p ip2-q2)q
* Phil. Mag. vol. xxxiv. no. 202, p. 260.
298 Mr. L. C. Jackson on
and these values put in (9) and (10) give the special
solution
(m? â€” q2)u . . . (p2 â€” m2)u . \
yâ€” i Â« -A sinÂ»H-V-= wv~ sin at,
(pâ€”q2)p r (p2-Q2)q I 0)
â€” (p2â€” m2)(m2 â€” q2)u . (m2 â€” q2)(p2 â€” m2)u .
z â€” â€” - 2 , / -2, - sin pt + 2/2 2^ - sm Qt '>
m2u(p~ â€” q'i)p nroi(p'i â€” qi)q *
so
â– -f=f=j2* and Â° =i, . . . (21)
where G and H are the quick and slow s vibrations.
(ii.) Upper bob struck.
Here we may write
These conditions put in (9) to (12) give equations
satisfied by
e=0, = 0, t=^^y F.-^p^. . (23)
So inserting these values in (9) and (10) we have the
special solution
â€” m2uv . m2a.v . /,,,N
-m>Â«npf+(^-fivÂ«n0i . . (25)
so
(f-q2)p K {p2-q2)q
E â€”Q j G ( p2 â€” m2) q
F p H (m2-qJ)p
Note the contrast between (21) and (26).
(26)
(iii.) Lower bob displaced ; upper free.
Let the displacement a of the lower bob M be produced
by a horizontal force. Then the corresponding displace-
ment z of N when at rest can be found statically. We thus
obtain
Â»â€” ' >=W t = Â°> gpOfar.-O.;. (27)
Variably Coupled Vibrations. 299
These conditions inserted in (9) to (12) give equations
satisfied by
- *" V _ P*~Â£ + (>2-g2)a2-(2a3 + *)m2 1
(i+Â«2)(p2-^) V"' L (28)
tt (2aÂ» + l)(mÂ»Â«)
* r ~ (l + a^p'-qr
These values put in (9) and (10) give the special
solution :
p2-q2 + (p2-q2)cc2-(2cc^u)m2
, (2a2+l)(7n2a) # ,0Q.
+ (>^^Â«^^ ' â€¢ (29>
-(p2-m2){p2-q2+(p2-q2)a2-(2c-g>+(p'--g8)Â«,-(2Â«, + a)m2}
H""~ (2a* + l)(msa)
(iv.) Upper bob displaced ; lower bob free.
This case may be represented by
y= ah, z = b, dÂ£ = 0, ^ = 0 for * = 0. . (33)
These put in (9) to (12) give equations satisfied by
T , t r, â€”a.q2b v ocp2b
.-5, *=j, E=^-?, F = ?^?. . (34)
300 Mr. L. C. Jackson on
These values in (9) and (10) give the special solution
â€” otq2b , ap2b
y = (f^) C0SPt+f^f cos $>
Y â€¢ (35)
(p2-m2)q2b (m*-q*)p*b
z=(2 2fS-C0S^ + 7~2 -A 9 cos at.
(p2 â€” q2)m2 2 (p2~qz)m2 * J
So
F />Â» and K-{m2-q*)p>' ' " * (dbJ
III. Relations among the Variables.
Fig. 2 is a graph showing the relation between y and a,
the couplings being ordinates and the values of a abscissae.
The data for the graph are given in the following
table : â€”
Coupling.
per cent.
0
5
10
Values of a.
0
005006
0-1005
Frequency ratio p
1-000
1051
1-106
15
01517
1-121
20
0-2041
1-226
25
0-2582
1-294
30
0-3145
1-362
35
0-3737
1-475
40
0-4364
1-542
45
0-5039
- 1-656
50
0-5780
1-767
55
0-6586
1-910
60
0-7499
2-083
65
0-8553
2-296
70
0-9802
2-572
It will be observed that, while the general solutions for
the present system are the same as those for the Cord
and Lath Pendulums, the equations for the couplings are
not the same, there being no term in a in the equation for
the gravity-elastic arrangement.
Fig. 3 is a graph showing the relation between y
and -.
9
Variably Coupled Vibration
s.
3(
775
70
Fig. 2.
65
60
55
/
50
45
/
40
35
30
25
20
15
10
5
0 0
1 0
â– 2 0
3 0
â€¢4 0
â€¢5 0
6 C
â€¢7
0
8 0
â€¢9 i
OL
IV. Experimental Arrangement and Results.
The actual experimental arrangement used is illustrated in
fig. 4. It consists of a lath L clamped at A and carrying a
bob X. The gravity pendulum M is attached to L at P,
P being movable. The bob M is of special construction.
It enables a spot of light to be focussed on a naked photo-
graphic plate B, which is moved by hand in a frame between
302
Mr. L. 0. Jackson on
guides perpendicularly to the motion of M, the room in which
experiment is made being dark. The box part of: M contains
an electric battery which lights a small lamp inside C. The
light then passes through a pinhole and is focussed in B by
a lens in D. By this means a trace is obtained directly for
the motion of M. If the apparatus is to be used for demon-
stration purposes for a class of students, the bob M is replaced
by a funnel and sand, as in arrangements used in papers
previously mentioned.
Variably Coupled Vibrations.
Length of lath pendulum = 100 ^ cms.
Length of gravity pendulum = 68*5 cms
303
Fio-s. 1-32 (Pis. VI. & VII.) are photographic repro-
ductions of traces obtained for the motion of the lower bob
under various conditions.
Figs. 1-16 (PL VI.) are traces obtained when the upper
bob was struck, the coupling ranging from 5 per cent,
to 65 per cent. The first figures, i. e. those for the smaller
couplings, show very well the phenomenon of beats and the
slow surging to and fro of the energy of the bobs. The
series shows in a marked manner the effect of a progressive
tio"htenino- of the coupling until, in fig. 14, the case of the
note and its octave (frequencies p : q = 2 : 1) has been reached
very nearly. It will be seen that a coupling intermediate
between that of fig. 14 (58 per cent ) and that of fig. 1 5
(62 per cent.) would give the ratio p:q = 2:l. By com-
parison with the table in Section III., it will be seen that
304 Variably Coupled Vibrations.
the experimental result is in good agreement with the theory.
Fig. 16 shows the effect of frequencies nearly 5:2; theory
indicates that for 65 per cent, coupling Â£>: q = 2'Â± : 1.
Figs. 17-24 (Pl.YII.) are traces obtained when the upper
bob was displaced, the lower bob hanging free. A com-
parison with PL VI. brings out the dependence of the details
of the traces on the initial conditions. Thus fig. 20 (58 per
cent.) is a trace for the ratio p : q nearly equal to 2 : 1 ; but
the characteristic " kink " of the 2 : 1 curve is hardly visible,
low down in the troughs of the curve. Fig. 21 (58 per
cent.), on the other hand, shows the "kink" very well ; but
the trace is slightly distorted because the bob possessed, at
the time, a small transverse motion as well as the correct
longitudinal motion. Fig. 22 (approx. 59 per cent.) shows
the effect of a combination of frequencies rather greater than
2 : 1, producing a wandering of the "' kink," with a definite
period up and down the main curve. The figure shows
rather more than a complete cycle of this wandering.
Figs. 25-32 (PL VII.) are traces obtained when the lower
bob was displaced, the upper bob being free. This was effected
by a horizontal thread, which was burnt when all was steady.
The coupled system here described thus presents a fairly
close mechanical analogy to the case of coupled electric
circuits, as will be seen by the foregoing theory and experi-
ment. On account of the simplicity of the arrangement
it is a convenient model by the aid of which the somewhat
abstruse subject of coupled electrical vibrations can be
demonstrated to a class where visible results are needed
to satisfy the non-mathematical student.
Summary.
1. In the present paper the mathematical theory of a
coupled system consisting of a gravity and an elastic
pendulum is developed and confirmed experimentally.
2. The paper is illustrated by 32 photographic repro-
ductions of the traces obtained for the motion of the lower
bob under various conditions of starting and coupling.
3. The system here discussed gives very similar results to
those previously obtained with the Cord and Lath Pendulums
by Barton and Browning, and can be used as an analogy to the
electrical case of circuits of equal inductances and frequencies.
In this mechanical case, as in the electrical one, the motions
of the components of the systems are not interchangeable.
4. It is hoped later to deal with the more general case of
the same arrangement in which both masses and periods are
unequal.
Physical Department,
University College, Nottingham,
June 1919.
[ 305 ]
XXIX. On the Magnetic Susceptibilities of Hydrogen and j
some other Gases. By Take Soke *.
Index to Sections.
1. Introduction.
2. Method of measurement.
3. Apparatus for measurement.
(rt) Magnetic balance.
(b) Compressor and measuring tube.
4. Procedure for measurements.
(a) Adjustment of the measuring' tube.
{b) Determination of the mass.
(c) Method of tilling- the measuring tube with gas.
(d) Electromagnet.
(e) Method of experiments.
5. Air.
6. Oxygen.
7. Carbon dioxide.
8. Nitrogen.
9. Hydrogen.
(a) Preparation of pure hydrogen gas.
(b) Filling the measuring tube with the gas.
(c) Results of magnetic measurement.
( d) Purity of the hydrogen gas.
10. Concluding remarks.
Â§ 1. Introduction.
IN the electron theory of magnetism, it is assumed that
the magnetism is due to electrons revolving about the
positive nucleus in the atom ; and hence the electronic
structure of the atom lias a very important bearing on its
magnetic properties. The models of the atoms or molecules
hitherto proposed are so constructed as to explain only the
phenomena of light ; but the question whether the nature
of atomic or molecular magnetism, due to the system of the
revolving electrons, agrees with the results of observation
or not, is in most cases not touched at all. For example,
Bohr's model t of hydrogen molecules explains very satis-
factorily the light dispersion of hydrogen ; but its magnetic
polarity is paramagnetic in contradiction to the observed
fact that hydrogen gas is diamagnetic. A correct theory of
the constitution of the atoms or molecules must, however,
not only explain the phenomena of light, but also their
magnetism. In this respect, a knowledge of the magnetic
susceptibility of various gases, especially those of hydrogen
* Communicated by the Author.
t N. Bohr, Phil. Mag. xxvi. p. 857 (1913); P. Debye, Site, d. math.-
phys. Klas. d. Akad. d. Wissensch. Munchen, p. 1 (1915).
Phil. Mag. S. 6. Vol. 39. No. 231. March 1920. X
306 Mr. Take Sone on the Magnetic Susceptibilities
and helium, is very important. In spite of this fact, owing
to the great difficulty in measuring the susceptibility of
gases, only a few cases â€” oxygen and air â€” are known, in
which the value of susceptibility can be given with fair
accuracy. For other gases the values of susceptibility by
different observers show large discrepancies not only in
magnitude, but sometimes in sign. Hence an exact and
more extended determination of the magnetic susceptibility
of different gases was thought to be desirable. The present
investigation was undertaken at the suggestion of Professor
K. Honda about two years ago and is still in progress.
On the other hand, the theory of magnetization of the
gases has been successfully developed by P. Langevin * ;.,
the conclusions arrived at agree in many points with the
observed facts ; but there are many others which cannot be
explained by his theory. According to him, the molecules
of a paramagnetic gas have each a definite magnetic moment^
which is comparable with that of a ferromagnetic substance,
but those of a diamagnetic gas have no magnetic moments,
so that there is a fundamental distinction between para-
magnetic and diamagnetic substances. The diamagnetism
is of an atomic nature and therefore cannot vary with
temperature, or by the change of states, or by any mode of
chemical combination. These conclusions do not, however,
accord with the fact that the susceptibility-atomic weight
curve for elements f changes quite continuously in passing
from the paramagnetic elements to the diamagnetic, and
that the susceptibility of tin J changes its sign at the trans-
formation point and during melting. Professor K. Honda Â§
modified Langevin's theory of paramagnetic and diamagnetic
gases so as to include the magnetization of liquid and solid
states, and gave a different aspect to the distinction between
paramagnetic and diamagnetic substances. According to
him the observed susceptibility % is the sum of the Langevin
paramagnetic and diamagnetic susceptibilities % and %d ;,
that is
which may be paramagnetic or diamagnetic, according as
Since xp depends on the configuration of the atoms in a
* P. Langevin, Ann. de cliim. et de phys. viii. p. 70 (1905).
t K. Honda, Ann. d. Phys. xxxii. p. 1027 (1910).
X K. Honda, he. tit.
Â§ K. Honda, Sci. Rep. iii. p. 171 (1914).
of Hydrogen and some other Gases. 307
molecule, it may change with temperature, or by the change
of states, etc. ; and hence the observed susceptibility ^ may
change in a similar way, as actually observed. The con-
tinuous change of the susceptibility-atomic weight curve
from the paramagnetic elements to diamagnetic above
referred to is also explained on the same basis.
According to the above theory, the molecules of a para-
magnetic substance must therefore possess a definite magnetic
moment, while those of a diamagnetic substance have only
a small magnetic moment or none. In Bohr's model of a
hydrogen molecule, v is decidedly greater than ^*, and
therefore % or XP + Xd *s positive in contradiction to the
observed fact.
In a recent paper, Professors Honda and Okubo f proposed
a new theory of magnetization of the gases. According to
the kinetic theory of gases, besides translational motions,
â€¢ the molecules of a gas are continuously making rotational
motions about their centres of mass ; and in their theory,
these molecules are treated as gyroscopes. Since the axis of
rotation of the molecules does not in general coincide with
the magnetic axis, the magnetic moment of the molecules is
supposed to be resolved in the direction of the axis of
rotation and that perpendicular to it. Under the action of
a magnetic field, the paramagnetic polarization results from
the former component and the diamagnetic polarization
from the latter, and therefore a resultant polarization is the
sum of these two. The resultant may be positive or negative,
according as
paramagnetic polarization ^ diamagnetic polarization.
The theory proves that the sign of the magnetization of a
gas depends on the shape of the molecules, and not in the
least on their magnetic moment. In fact, a gas whose
molecules have a definite magnetic moment comparable with
those of iron may be diamagnetic, provided the axis of
rotation is perpendicular to the magnetic axis. This kind
of diamagnetism is not dealt with in any of the previous
theories. In order to test these theories it is necessary to
have the correct values of susceptibility of different gases,
which are at present scarcely known.
One of the chief difficulties which we encounter in the
determination of the magnetic susceptibility of gases, lies in
the preparation of pure gases, i. e. those which are perfectly
* J. Kunz, Phys. Rev. xii. p. 59 (1918).
t K. Honda and J. Okubo, Sci. Rep. vii. p. 141 (1918) ; Phys. Rev.
viii. p. "6 (1919).
X2
308 Mr. Take Sone on the Magnetic Susceptibilities
free from air, and the other in the extreme smallness of the
volume susceptibility of gases. In the present research I
paid special attention to tbe preparation of pure gases, the
removal of the air contained in the generators and purifiers
being the constant object of my endeavour. As for the
measurement of the magnetic susceptibility, I succeeded in
overcoming the difficulty by constructing an apparatus, by
means of which I could seal gases in a glass tube at a very
high pressure without the least fear of leakage. This
apparatus enabled me to use in each case a quantity of gas
sufficient for the determination of its magnetic susceptibility
and density. A special magnetic balance of high sensibility
was constructed for measuring the magnetic force acting on
the gas which was sealed in the glass tube and placed in a
strong magnetic field. The details of the method and the
arrangement of the experiments are given in the following
pages.
Â§ 2. Method of Measurement.
The method of measurement is based on the fol lowing
principle : â€” A cylindrical rod made of the material to be
tested is vertically suspended between the horizontal pole-
pieces of an electromagnet from an arm of a magnetic
balance specially constructed, the
lower end of the rod being placed in Fig. 1.
the axial line of the pole-pieces and -i â€” /\ â€”
the upper end in a place where the
magnetic field is negligibly small. n
Suppose at first the balance to
be in equilibrium, with no mag- \
netic field acting on it, by applying
the field the rod is supposed to
undergo a slight upward displace-
ment 8a, but in equilibrium acted y'
on by a force / arising from an
inclination of the balance beam.
Then this force is just equal to the lifting force due to the
magnetic field H. The work done by the force is equal to
the change of the magnetic energy of the rod, that is
f8a=^W^8a,
where S is the cross-section of the rod, k and k are the
susceptibilities of the test specimen and the surrounding
medium respectively.
of Hydrogen and some other Gases. 309
Hence /= â€” ~ â€” IPS,
or k â€” k =
2
H2S*
The force /is measured by a small deflexion of the balance
beam, which causes a vertical rotation of a small mirror
suspended by a bi filar system, the upper ends of the fibres
being attached to the lower end of the pointer and to a fixed
stand. This rotation of the mirror is measured as usual by
a scale and telescope. The above method is due to Lord
Kelvin.
For a very small displacement of the suspended tube the
force is proportional to 8a, and consequently to the deflexion
of the scale $ ; hence c being a proportional constant^,
we get
f=cS.
Introducing this relation into the expression for kâ€”k\
we get
,_ 2cB
K *""H*S'
If the intensity of the field remains constant in the range
2c
of the displacement, the factor r=â€”^ is also constant. Let p
denote this factor, then the above expression becomes
kâ€” k â€” pS.
In the present experiment the measurements of the sus-
ceptibilities of the gases were always made relatively to
water or to air ; that is, for the case of air, the comparison
was made with distilled water, while for other gases the
comparison was always made with dry air. In the present
day the susceptibility of pure water* is accurately known,
its value â€” O720 xlO-6 was assumed in the present experi-
ment. The susceptibility of air was determined relatively
to water.
The upper half of a glass tube separated by a glass
partition in the middle was filled with the gas" or liquid
under examination, while the air in the lower half of the
tube was evacuated and its lower end sealed. The tube was
then vertically suspended from the arm of the balance
* P. Seve, Jour, d, Phys. (5) iii. p. 8 (1913); de Haas u. Drapier,
Ann. d. Phys. xlii. p. 673 (1913) ; A. Piccard, Arch, de Geneve, xxxv.
p. 209 (1913).
Fiff. 2.
310 Mr. Take Sone on the Magnetic Susceptibilities
between the pole-pieces of a Weiss electromagnet, the upper
surface of the partition being placed on the axial line of the
pole-pieces (fig. 2). The field was then applied and the
corresponding deflexion of the
scale observed. Since the tube
alone produced some deflexion
of the scale it was necessary to
eliminate the effect by making-
two similar observations, first
after evacuating the upper half,
and secondly after filling it with
distilled water.
If the virtual susceptibility of
the system below the glass par-
tition be denoted by k!, we have
the following relations for the
three cases, when the upper half of the tube is evacuated,
and when it is filled with the gas to be tested and with water
respectively :
K0 â€” /c'=pS ,
/egâ€”/e'=p8g,
Kw â€” K'â€”phw,
where k0, teg, and kw denote the susceptibilities of the empty
space, the gas, and water respectively, and 6\>, 8g, and 8tv are
the corresponding deflexions of the scale.
Eliminating k! from these equations and putting k equal
to zero, we get the following relation between the suscepti-
bilities of the gas and water :
8W â€” 8C
In the actual case, since the glass tube is not placed in a
perfectly symmetrical position with respect to the axial line
of the pole-pieces, the term k0 is not zero though it is a
small quantity, and it is the susceptibility due to the glass tube
itself. In the terms 8â€ž and Sâ€ž, the above quantity k0 due to
the glass
and 8,
tube is involved, and the differences 8g 80 = dr/
8Q = dw are the true deflexions due to the glass and
water respectively. And finally we get
K.g __ dg
â€¢Cm CI m
If the densities of the gas and water be respectively pg
of Hydrogen and some other Gases.
311
and pWi we have for the ratio of tlie specific susceptibilities
X and Xv Â°^ these two substances
2k
Xw
or
dJpw
_ dg/mff
du-lmtv
where mg and mw are the total masses of the gas and water
occupying the same volume of the tube respectively. The
same formula may also be used wrhen water is replaced by
air, in which case we obtain the susceptibility of a gas
relative to air.
Â§ 3. Apparatus for Measurement.
(a) Magnetic Balance.
The magnetic balance and its accessories used in measuring
the magnetic susceptibilities of several gases are diagram-
matically shown in fig. 3.
AB is an aluminium arm of the magnetic balance, 80 cm.
long, C an agate knife-edge resting on a smooth plane of
steel, and HD an aluminium pointer, a similar counterpoise
being attached to the same arm, but on the opposite side of
the beam. DE shows the side view of the bifilar system
312
Mr. Take Sone on the Magnetic Susceptibilities
Eigr. 4.
consisting of two Wollastone wires 0*015 mm. thick, M is a
small mirror attached to the lower end of: the bifilar system
and facing at right angles to the plane of the figure, and F
a copper vane damper dipped in vessel containing a mixture
of petroleum and machine oil. I is a rectangular brass
pillar with a slide-arm carrying a fixed suspension of the
bifilar system. Q is a Quincke microscope with an ocular
micrometer whose smallest division corresponds to 1/G0 mm.
With the microscope the breadth of the bifilar on its upper
end can be observed. S is a trifilar system consisting of
two horizontal Y-shaped wooden frames with a small
aluminium Y between them, these frames being connected by
three fine copper wires, as shown in fig. 4. The aluminium
frame can be moved upwards or downwards,
so that one can make a minute adjustment
of the height of the suspended tube. T is
the measuring tube suspended between the
pole-pieces of a Weiss electromagnet of inter-
mediate size, t is a thermometer placed near
the measuring tube to determine the tempe-
rature of the specimen under examination.
P is a microscope (shown in section) readable
to 0*005 mm., with which one can adjust the
measuring tube to a correct position. K is a
collimator tube of a spectroscope used for the
purpose of clamping the arm of the balance
by means of its vertical slit. This is necessary
when it is required to take the measuring-
tube away from the arm, or when the tube is
to be adjusted to a correct position by means of the trifilar
system without giving the least disturbance to the balance.
L is a pan in which a balancing weight is to be placed,
and G a metal damper dipped in a vessel of machine oil.
W is a glass cup containing water, into which is dipped a
fine glass tubing, connected with a fine copper tube U, both
being filled with water.
The whole arrangement was set upon a stone foundation
and covered with a case to prevent the disturbances due to
air currents. The case has a large glass window, through
which we could observe the deflexion of the mirror with a
scale and telescope. To make the finer adjustment of the
orientation of the mirror from outside, an arrangement shown
in fig. 5 was used.
AB is a kind of reservoir made of glass filled with water
and mercury; the left end of this reservoir is connected
with the copper tube U described above and the other end
of Hydrogen and some other Gases. 313
with a fine capillary tube C, O'l mm. in diameter through a
three-way-cock. One of the three ways in connected with
a mercury reservoir D, and to the left end of the capillary
IS1. 5.
tube, a calcium chloride tube with a long rubber tubing is
connected. By this arrangement we can pour into or draw
out any desired quantity of water from the glass cup W
shown in fig. 3, by forcing the air or sucking it at the end
of the rubber tubing F. The reservoir D serves in the case,
when a large quantity of water is desired to be supplied or
extracted from the cup.
(b) Compressor and Measuring Tirf>e.
The compression of the gas was made with a Cailletet
hydraulic compressor of an ordinary type. The compressing
cylinder was replaced by a cast-iron cylinder, specially
designed for the present purpose and having a capacity of
1300 c.c. The pressure-gauge was also replaced by another
capable of measuring up to 70 atmospheres and graduated
to one atmosphere.
The glass tube, in which the gases are to be filled at a
high pressure, has the following construction (fig. 6) : â€”
'ihe mechanism of the valve of the measuring tube is
similar to that of the pressure-gauge commonly used for
compressed-gas-bombs. A is a hollow brass cylinder having
two bores in it, a narrow straight hole, 1 mm. wide, is bored
through from the left end to the central hole terminating in
a cone. In this hole a small brass cylinder P lightly fitted
to the hole is pressed on a rubber plate on the right end of the
hole by a weak spring. On the left end of this cylinder Pr
a small piece of ebonite is imbedded. The right opening of
the cylinder A is provided with a cup C, which can be
screwed into the cylinder, till it firmly presses the rubber
plate covering the right opening of the central hole with a
thin metal ring. Within the cup 0, there is a piece of
314
Mr. Take Sone on the Magnetic Susceptibilities
metal Q which can be moved smoothly along the axis of the
cup by means of: a screw E. To the left end of the cylinder
A, a brass piece B, and to its lateral surface, another piece D
is screwed in. The former piece is tightly connected with
the measuring tube, and the latter with the glass tube
forming the projecting neck of the compressing cylinder.
Fig. 6.
fW
(Scale 1:1.)
This connexion between the glass tubes and the brass pieces
is made airtight by sealing-wax mixed with a small quantity
of linseed oil. In order to prevent the flowing of wax into
the interior of the tube, when melted wax is poured into the
interspace between the brass piece and glass tube, a thin
rubber plate is placed at the end of the glass tube. All the
packings for the screws are made of thick lead plates.
If the measuring tube with the brass pieces are connected
with the compressing cylinder by means of the screw D and
the inside air is evacuated by a pump connected with the
compressing cylinder, the piston P is displaced to the right
by a spring, so that the interior of the glass tube and of the
cylinder A communicate with each other. If after the gas
is compressed into the measuring tube, the screw E is turned
and the piston P made to press on the rubber plate, the piston
tightly closes the left hole of the cylinder A. The screw D
is then disconnected, and now the measuring tube with the
compressed gas inside can be suspended from the beam of
the magnetic balance between the pole-pieces of the electro-
magnet for the determination of the susceptibility of the
of Hydrogen and some other Gases. 315
If the volume oÂ£ the compressed gas is to be measured,
a screw-head similar to D with a short piece of glass tube is
screwed in the side hole ; this tube is then placed under a
receiving vessel for the gas. By turning the screw E very
slowly, the compressed gas can be let into the receiving-
vessel.
Â§ 4. Procedure for Experiments.
(a) Adjustment of ilie Measuring Tube.
The position of the measuring tube suspended from the
beam of the balance was accurately determined by means of
the microscope. The forward or backward displacement of
the tube from the central line joining the centres of the
pole-pieces was adjusted with an accuracy of O'l mm. by aid
of the lines marked on the pole-pieces. The height of the
tube could easily and quickly be adjusted by means of the
trifilar suspension system, with an accuracy of 0*05 mm.
The breadth of bifilar carrying the mirror was observed
with a Quincke microscope and adjusted with an accuracy
of 0*01 mm. The distance of the mirror from the scale and
telescope was 190 cm. When the breadth of the bifilar
suspension was 0'3 mm. the deflexion of one scale-division
corresponded to a vertical displacement of the measuring-
tube by about 1 X 10 "3 mm.
(b) Determination of the Mass.
The mass of the gases subjected to the experiment was de-
termined by two different methods and the results compared
w7ith each other. The first method was to determine the mass
directly by weighing, that is, from the difference of weights,
first when the tube was filled with the compressed gas under
examination, and afterwards when it was evacuated. The
evacuation was always made after the tube was repeatedly
filled and evacuated several times with pure hydrogen by
means of a Gaede auxiliary pump, and then the weighing
wTas conducted with the utmost care. The second method
was to determine the mass by measuring the volume of the
gas collected in an eudiometer. Knowing the pressure and
temperature of the gas, we calculated the volume of the gas
at the standard pressure and temperature, and by multiplying
the density of the gas at the standard conditions by the
volume thus obtained, the mass of the gas was obtained.
These two methods were used in most of the experiments and
the results were found to agree very satisfactorily with each
other. But, for the reduction of the observed results, either
316 Mr. Take Sone on the Magnetic Suscejrtibilities
of these results was used according to the kind oÂ£ gases
under examination. Thus the mass of carbon dioxide was
always determined by the weighing method, and that of
hydrogen chiefly by the volumetric method. The mass of
air sealed in the measuring tube at the atmospheric pressure
and temperature was obtained by calculation, the conditions
of the atmospheric air and the volume of the tube in which
it was sealed being; known. The volume of the measuring
tube was accurately determined by filling it with mercury.
(c) Method of filling the Measuring Tube with Gas.
The measuring tube is connected with the neck of the
compressing cylinder E by means of a screw D (fig. 7).
Fig. 7.
The lower end of the glass cylinder E is bent upward, and
connected by a short rubber tube with a glass tube coming
from a three-way-cock T. One of the three ways com-
municates with a Gaede auxiliary pump and another with
the gas generator or reservoir. The lower part of the
cylinder E is dipped in a mercury bath as shown in fig. 7 .
By means of the three-way-cock, the cylinder is first con-
nected with the pump, evacuated, and then the cock is
turned, the gas is introduced into the cylinder. Next the
cylinder is again evacuated and the gas introduced ; these
processes are usually repeated four or five times. Then the
air previously contained in the cylinder is removed, and
the cylinder now contains the pure gas at a pressure of abou ^
one atmosphere Then the cylinder E and the glass tube i
disconnected under the surface of the mercury in the bath^
oi Hydrogen and some other Gases. 317
A quantity of mercury then flows into the cylinder and
partially fills the bottom of it. In this state the cylinder is
transportable to the Cailletet compressor without any risk
of introducing other oases into it.
(d) Electromagnet.
The magnetic field was obtained by a Weiss electromagnet,
the pole-pieces being always 1 cm. apart. The end surface
of the pole-pieces was a circular section of 1 cm. in diameter.
The magnetizing current of 10 amperes produced a field of
22,000 gauss in the place where the magnetic measurement
was to be made. In the case of air 4 amperes were used
and the corresponding field 12,500 gauss. During the
magnetization a slight convection of air was produced by
the heating of the electromagnet, and this made the obser-
vation of the deflexion of the mirror somewhat difficult.
To avoid the disturbing effect the coil of the magnet with
Â© . Â©
the exception of the pole-pieces was entirely covered with a
winding of lead tubing and the water mantles, water being
constantly circulated through the lead tubing and mantles
during the observations.
The intensity of the field was measured by means of an
exploring coil and a ballistic galvanometer as usual.
(e) Method of Experiments.
In the magnetic balance a delicate knife-edge rests on a
smooth steel plane, so that a very minute gradual displace-
ment of the knife-edge, either translational or rotational,
can never be absolutely avoided. This gradual displacement
is usually accelerated when the field was repeatedly applied,
causing the gradual displacement of the zero point on the
scale. However, by comparing the results obtained when
such a gradual displacement of the zero-point occurred and
when it was absent, it was found that this displacement did
not affect the final results, provided the mean of the successive
zero-points in each observation be taken as the true zero of
the deflexion.
In making the observations, we first passed a current of
10 amperes in the electromagnet, and the maximum deflexion
of the scale was observed. It took usually 30 to 60 seconds.
Then the current was quickly reduced to zero, and the final
deflexion or the zero-point was observed. These processes
were repeated usually ten times, and the mean of these
deflexions was taken. The whole process required about
ten minutes. The temperature of the gas under examination
318 Mr. Take Sone'ow the Magnetic Susceptibility
was carefully observed at each set of observations with a
thermometer suspended near the measuring tube in the
space between the poles of the electromagnet. A current
of water was constantly passed around the electromagnet
during the observation.
Â§ 5. Aie.
The magnetic susceptibility of the atmospheric air has
been determined by many investigators, such as Faraday,
Becquerel, Quincke, and more recently by Curie. But other
investigators such as Du Bois, Hennig, and Piccard, have
deduced the susceptibility of air from that of oxygen by
neglecting the susceptibilities of nitrogen and other gases
present in the atmosphere.
In the following table, the values of the volume suscepti-
bility of air and also the ratio of the susceptibilities of air
and oxygen in four cases, in which the susceptibilities were
independently determined, are given : â€”
Table I.
Date. Observer. Ka . W\ t. KJK0>-
1853 Faraday* 0'024 ... 0-184
1855 Becquerel t 0-025 ... 0"208
1888 Quincke \ 0-032 16Â° 0*248
1895 Curie Â§ 0-027 20Â° 0'232
1888 Du Bois || 0-024 15Â°
1893 Hennig Â«[ 0-024 25Â°
1913 Piccard-* 0"029 20Â°
* M. Faradav, Exp. Res. of Elec. (3) p. 502.
+ E. Becquerel, Ann. de Chim. et de Phys. (3) xliv. p. 223 (1855).
X Gt. Quincke, Wied. Ann. xxxiv. p. 401 (1888).
Â§ P. Curie, Journ. de Phys. iv. p. 197 (1895).
H H. Du Bois, Wied. Ann. xxxv. p. 137 (1888).
5] R. Hennig-, Wied. Ann. 1. p. 485 (1893).
** A. Piccard, Arch, de Geneve, xxxv. p. 458 (1913).
Tn the above table we see that the magnetic suscepti-
bility of air as obtained by several observers shows a large
discrepancy, and also the ratio of the susceptibilities of air
and oxygen directly determined is not constant. Hence
it is to be concluded that the susceptibility of air and its
relation to that of oxygen are not yet correctly known.
In the present research the susceptibility of the atmo-
spheric air was determined relatively to that of redistilled
water. The air to be examined was introduced into the
compressing cylinder after passing through the tubes con-
taining solid potassium hydroxide, calcium chloride, phos-
of Hydrogen and some other Gases. 319
phorns pentoxide^ and cotton ; thus the air in the cylinder
was entirely free from carbon dioxide, moisture, and dust.
The air was then compressed into the measuring tube at a
pressure of about 30 atmospheres and subjected to the
magnetic measurement. The masses of the gas and the
water were determined by weighing. In this case the
magnetizing current was usually 4 amperes, and the breadth
of the bifilar 1 mm. ; bat by changing the bifilar distance
and applying a weaker or stronger field, a moderate deflexion
of the scale could be obtained. In the following table one
set of observations is shown as an example : â€”
Table
II.
Sept. 5, 1918
Air (p=30 atm.)
Vacuum.
lh 50â„¢
p.m., /=24Â°-5C
.,6=60 0. 2h35mp.M.
,*=24Â°-8C.,
6=60-0.
m
B+Â«*o=89-0927
gm.
mo
=88-9438 gm.
C.
S.
8.
C.
S.
8.
amp.
cm.
cm.
amp.
cm.
cm.
0
56-90
17-55
0
39-40
0-15
4
74-45
17-79
4
39-55
0-35
0
5675
17-55
0
39-20
0-20
4
74-30
17-60
4
39-40
0-20
0
5670
17-60
0
39-20
0-20
4
74-30
17-70
4
39-40
0-24
0
56-60
17-60
0
3916
0-24
4
74-20
17-70
4
39-46
0-20
0
56-50
17-62 (mean)
0
39-20
0-22 (mean)
Water. Calculation.
3h 35m p.m., ^=24Â°-2 C, 6=60-0, da=17G2-0'22=lT40 cm.,
- mw+Â»Â»0=93-1178 gm. <â€ž= -14-64-0-22= -14-86 cm..
C. S. f.
amp. cm. cm.
0 61-29 -14 59 Xa =
ma=0-1489 gm., mn =4-1740 gm.,
17-40 K
0-1489
4 46-70 -14-70
0 61-40 -14-69 -I4 86*
4 40-71 -14-79 *" 4-1740
0 61-42 -14-56 Xa_im
4 46-86 -14 56 ^ ~ 3^562"
0 61-42 -14-62
4
X,
4680 -14-70 Xa=23-60xl0-Â«.
0 61-50 -14-64 (mean)
In the above table h is the breadth of the bifilar expressed
in the scale division in the ocular micrometer, 60 divisions
corresponding to 1 mm. mo2, m0, and mw are the masses
of oxygen, the tube evacuated, and the water respectively..
320 Mr. Take Sone on the Magnetic Susceptibilities
do2 and dw are the deflexions oÂ£ the scale due to oxygen and
water respectively. C is the magnetizing current, S the
scale reading, and S the deflexion of the scale. K is a
constant which depends on the sensibility of the apparatus
and the intensity of the field.
Fifteen sets of such observations were made at a mean
temperature of about 25Â° 0. ; reducing these observations to
20Â° C. by assuming Curie's law, we obtain the following-
result : â€”
106.%Â« = 23*95, 23^52, 24*20, 23*58, 23*58, 24*15,
23 69, 24*04, 23-65, 24*37, 23*81, 23*42,
24-06, 23*66, and 24-13,
the mean value is then
%a = 23-85xl0-6at20Â°C.
with a mean error of +0*07 x 10~6.
Multiplying the density of air at 20Â° C. and 760 mm.
pressure, we get as the magnetic susceptibility per unit
volume of dry air at 20Â° 0. and at the normal pressure,
/^=Q*02373xl0-6 + 0-00009xl0-6.
Â§ 6. Oxygen.
Oxygen is the only gas whose magnetic susceptibility
has been determined with a fair accuracy ; yet the values
obtained by different observers differ so widely from each
other that the extreme values deviate from the mean by
more than 10 per cent.
The following table contains the values which have been
determined by several investigators : â€”
Table III.
Date. Observer. k'a . 106. t.
1853 Faraday* 0143
1855 Becquerelt 0*149
1888 Quincke + 0*129 16Â°
1888 DuBoisÂ§ 0-117 15Â°
1893 HennigJ 0*120 25Â°
1895 Curie Â«T 0*115 20Â°
1913 Piccard** 0*141 20Â°
1916 Booptt 0*146 16Â°
* M. Faraday, he. cit. \\ R. Ilemng, loc. cit.
t E. Becquerel, loc. cit. 51 P- Curie, loc. cit.
\ G. Quincke, loc. cit. **.* A. Piccard, loc. cit.
Â§ H. Du Bois, loc. cit.
tf W. P. Poop, Phys. Rev. vii. p. 529 (1916).
In the present experiments the pure oxygen was obtained
Ot
if Hydrogen and some other Gases. 321
by the decomposition of: a solution of potassium hydroxide
by electrolysis. The electrodes were of sheet nickel attached
to stout nickel rods, the strength ot the solution being two
normal. The cell consisted of two concentric glass cylinders,
the inner one having no bottom ; one of the electrodes was
placed in the inner cylinder and the other placed in the
space between these two cylinders. The current density was
0*05 ampere per sq. cm. The generated gas was collected in
a large glass reservoir by replacing water. Before intro-
ducing the gas into the measuring tube, she gas was purified
by passing through bottles and tubes containing strong
sulphuric acid, solid potassium hydroxide, and phosphorus
pentoxide.
The susceptibilities of the gas thus purified and the air
treated likewise, as described in the last Section, were
compared at the ordinary temperature and pressure ; the
ratio of the mass of the gases being determined by calcu-
lation by knowing the temperature and pressure of the atmo-
sphere at the time of filling the gases in the measuring tube.
In the following table an example of the results of the
measurement is shown : â€”
Table IY.
March 19, 1918.
Air.
Vacuum.
(*=l
2o-9C.,p=748-0mm.J *=13Â°-5C.)
7]1 15ul p.m
.,*=15Â°-5 C, 6=20-0.
71
' 45,n f.m.,-*=15Â°-3 C, 6=20-0.
0.
s. s.
C. S. 8.
amp.
cm. cm.
imp. cm. cm.
0
41-50 10-70
0 50-75 22-90
10
52-20 10-20
10 73-65 22-40
0
42-00 9-60
0 51-25 22-30
10
51-60 10-30
10 73-55 22-75
0
41-30 1010
0 50-80 22 60
10
51-40 10-40 ,
10 73-49 22-40
0
41-00 10-22 (mean)
0 51-00 22-56 (mean)
02.
Calculation.
=13o"l'0.,7Â»=748-0mmÂ«l *=13Â°-5
7h 55m p.m., *=14Â°-75 C.: 6=20-0
i.O.)
^=69-89-10-22 = 59-69 cm.,
O.
S. d.
^=22-56-10-22= 12-34 cm.,
amp.
cm. cm.
0
10
13-60 72-00
85-60 69-20
*â– a â–
59-67 0 001210 287"3
>0Â° C. ~ 12'34 0-001335 X 288-3
0
10
16-40 69-00
85-40 69-50
=4-365.
0
15-90 69-30
(Xo2
)20oc.-104-1^10~(-
10
85-20 69-80
0
15-40 70-15
10
85*55 70'15
0
15-40 69-89 (mean)
Jhil Mag.
. S. 6. Vol. 39. No. 231
. March 1920, Y
322 Mr. Take Sone on the Magnetic Susceptibilities
Eight sets of such observations made at various tem-
peratures ranging from 14Â° C. to 18Â° C. give the following-
values of the susceptibility of oxygen, which are all reduced
to the value at 20Â° C.
10*.Xo= 104*00, 104-48, 104*00, 104-10, 103*89, 104-12,
104*60, and 103*86, the mean of which is
Xo =104*1xlO-6at20Â°C..
the mean error being +0 09 x 10~6. Multiplying the density
of oxygen at 20Â° C. and 760 mm. pressure, we get as the
susceptibility of oxygen gas per cubic centimetre,
yrO2=0'1386 x 10"6Â±0-0001 x 10"G.
The above result is in fair agreement with Piccard's value,
which seems to be the most reliable among the values of
the previous investigators, but as he used a commercial
oxygen, the error which might arise from the impurity of
the gas wrould probably make the mean error of his result
larger than he had believed. He also determined the mass
of the gas by the absorption method, which is accompanied
by some uncertainty. In the present case the gas was
obtained by electrolysis and carefully purified, so that there
is no uncertainty about its purity. At any rate the error in
my case, if there is any, may arise from the determination
of the susceptibility.
If the magnetism of air is due solely to that of oxygen
present in it, the ratio of the specific susceptibilities of air
and oxygen should be identical with the ratio of the mass of
the oxygen in air to the total mass of air, that is, the ratio
-^ = f^ =0-2291
must be equal to 0*2315, the difference being 0*0024.
The difference amounts to about 1 per cent, of the total
value and is certainly beyond the experimental errors of the
measurements of the susceptibilities. The explanation for
this discrepancy will be given later on.
IÂ§ 7. Carbon dioxide.
About ten years ago Professor K. Honda * found an
anomalous behaviour of tin in that its magnetic property
* K. Honda, loc. cit.
of Hydrogen and some other Gases. 323
changes from a paramagnetic to a diamagnetic during
melting. Since that time no other substance showing a
similar change has been found. Meantime Mr. T. Xshiwara
in our laboratory found in the course of* his researches on
the magnetic susceptibility of chemical substances at low
temperatures, that solid carbon dioxide has a diamagnetic
susceptibility per unit mass of ^= â€” 0*42 x 10~6 in the
temperature range between â€” 100Â° C. and â€” 170Â° C.
In the literature we have only a few data for the magnetic
susceptibility of the gaseous carbon dioxide. The earlier
investigators, such as Faraday * and Becquerel f, agreed in
the view that the magnetism of gaseous carbon dioxide is too
weak to be detected by their experiments. Quincke { found,
however, that its specific susceptibility is %= +0*017 x 10"6;
more recently Bernstein Â§ found its volume susceptibility to
be k= +0-0002 x 10"6. If these two results be true, at
least in sign, then gaseous carbon dioxide is paramagnetic,
and we have, besides tin. one more example of a magnetically
anomalous substance. In this respect an exact determination
of the susceptibility of the gaseous carbon dioxide seemed
very interesting.
In the present experiment the carbon dioxide was
obtained by the reaction of dilute hydrochloric acid on
calcium carbonate. Pieces of pure marble previously boiled
in hot water for about 24 hours in order to drive off the air
occluded, were put in a Kipp apparatus with boiled distilled
water, care being taken not to expose the pieces to the air.
Xo air bubble was allowed to remain in the bottle. Then
the strong hydrochloric acid was poured into the apparatus
through the upper opening, and at the same time, by
expelling the water from the exit, we could easily replace
the water with hydrochloric acid of a moderate strength
without introducing any trace of air into the apparatus. By
this means we were able to obtain the carbon dioxide entirely
free from air. Before introducing the gas generated in the
Kipp apparatus into the compressing cylinder, it was first
passed through a bottle containing water, and then bottles
containing strong sulphuric acid and pieces of calcium
chloride. In the following table one set of observations is
given as an example.
*M. Faraday, toe. cit. t E. Becquerel, he. cit.
I q Quincke, loc. cit. Â§ Bernstein, Diss. Halle, 1909.
Y 2
324
Mr. Take So'ne on the Magnetic Susceptibilities
Table
V.
Jan. 13, 1917.
Vacuum.
CO
2(p = 2h
atm.).
lh 15m
p.m., Â£=17Â°
OC
., 6 = 20-0.
3h 15m p.m., Â£ = 19Â°
â€¢0 C, 6=20-0.
Â»io=89-3657
gm.
m
co2
-MÂ»0=8Â£
'5867 gm.
c.
S.
S.
C.
S.
&
amp
cm.
cm.
amp.
cm.
cm.
0
53-30
-8-50
0
65-60
-15-60
10
44-80
-9-20
10
50-00
-17-00
0
54-00
-7-90
. 0
6700
-15-40
10
46-10
-9-70
10
51-60
-1660
0
55-80
-8-80
0
68-20
-16-40
10
47-00
-8-90
10
51-80
-18-10
0
55-90
-7-90
0
6990
-13-50
10
48-00
-9-20
10
56-40
-15-50
0
57-20
-8-00
0
71-90
-17-90
10
49-20
-8-40
10
54-00
-18-00
0
57-60
-8-10
0
72-00
-15-90
1.0
4.9-50
-9-30
10
56-10
-15-90
0
58-80
-8-90
0
72-00
-16-31 (mean)
10
49-90
-9-70
0
59-60
-7-70
10
51-90
â€” 8'68 (mean)
Air (p=l atm.).
5b 40â„¢ p.m., ^ = 16o,0 C, 6 = 20-0.
mâ€ž+mn =89-3706 gm.
C.
amp.
0
10
0
10
0
cm.
53-70
54-50
53-70
54-60
53-70
8.
cm.
0-80
0-80
0-90
0-90
0-85 (mean'
Calculation.
dCOi>- -16-31+8-68= - 703 cm.,
. ^=0-854-8-68=9-53 cm.,
mco ; =0-2210 gm.,ma=0-0049gm.,
Co.
Xaz
-7-63 K
9-53 K
00049
= 1945 K,
"co2
xâ€ž
^ = -00177.
The values of the susceptibility obtained in three sets
of observations which were made at a mean temperature
18Â° C. are
lOG.%ca,= - 0-423, -0-429, and -0-416.
The mean value is
XCO2=-0-423xl0-e.
In the above calculation the susceptibility of air is corrected
for the temperature at the time of measurement. As the
of Hydrogen and some other Gases. 1^25
diamagnetic susceptibility is considered to be independent
of temperature, we can get by multiplying the density of
the gas at any temperature and pressure into the specific
susceptibility above obtained, the volume susceptibility of
the gas at that temperature and pressure ; thus we get as the
volume susceptibility of carbon dioxide at 20Â° 0. and 760 mm.
pressure,
Â«CO2=-0-000779xl0-6.
The same consideration is used in the calculation of the
susceptibilities of the diamagnetic gases investigated in the
present research.
The value of the specific susceptibility of carbon dioxide
obtained above is in fair agreement with the value
Xoo= -0-42 X 10-Â«
for the solid carbon dioxide obtained by Mr. T. Ishiwara
mentioned above. This sho ws that the specific susceptibilities
of carbon dioxide in the solid and gaseous states are almost
equal to each other. Putting for the moment this delicate
question of magnitude out of consideration, the close agree-
ment of these determinations seems to give strong confirm-
ation of the conclusion that the value obtained in the present
experiment for the gaseous carbon dioxide is not far from
the true value. At least we can assert that the magnetism
of carbon dioxide is diamagnetic, contrary to all previous
determinations. The paramagnetic result of the gaseous
carbon dioxide obtained by the previous investigators comes
probably from the impurity of the gas examined, such as a
trace of air mixed with the gas.
Of the three sets of measurements quoted above, the last
one was made by a method which was in some respects
different from the other two. The method employed was
the null-method described in the last section.
Namely, we measure the force / in terms of the volume
of water which is to be supplied or to be taken out of the
vessel hanging from the arm of the balance, in order to
bring the measuring tube to the initial position against the
magnetic force, or which is equivalent, to bring the deflexion
of the mirror to its initial reading on the scale. The volume
of water can be read from the volume of mercury thread in
the capillary tube.
The following table contains the data of the measurement
for carbon dioxide by means of the null-method. Here r is
the reading of the head of the mercury thread in the capillary
326 Mr. Take Sone on the Magnetic Susceptibilities
tube, 8 is the difference between these readings, and Bm the
mean for each set of observations : â€”
Table VI.
May 28, 1917.
C02 QÂ»=35 atm.)
Vacuum.
2=18Â°'5C., 6=10-0.
b=
:10-0, m0-
=98-7750 gm.
mCOo +m0 =99-0673 gm.
c.
r.
8.
*Â»â–
C.
r. 8.
.Â»i-
'.rap.
cm.
cm.
cm.
amp.
cm. cm.
cm.
0
27-32
-2-48
-2;65
0
29-00 -14-53
-1473
10
29-80
-2-82
10
43-53 -14-93
0
26-98
0
28-60
0
26-76
-2-34
-2-52
0
28-60 -14-55
-14-35
10
29-10
-2-70
10
43-15 -14-15
0
26-40
0
2900
0
26-25
-2-30
-2-55
0
29-35 â€”14-35
-14-63
10
0
28-55
25-75
-2-80
10
0
4370 -14-90
28-80
- 14-57(mean)
0
25;70
-2-45
-2-73
10
0
28-15
2515
-300
-2-61 (mean)
Air.
Calculation.
*=18Â°-0C, 6=10-0,
d
COo = - 14-57+2-61 = -11-96 cm.,
ma+7n =
=98-7795
gm.
dfl=7-02 + 2-61 =
= 963 cm.,
c.
r.
8.
*Â»â–
m
co =0-3223 gm.,
m =0-0045 gm.,
amp
cm.
cm.
cm.
2
0
10
24-57
17-70
6-87
7-05
6-96
Y -11-96 K
cÂ°i~ 0-3223
= -37-lK
0
24-75
9-63 K 01
Y = =isJ
"" 00045
l40K,
0
24-75
7-00
7-08
>
'co2 -37-1
10
17-75
7-15
Xa " 21"Â° "
0-0173.
0
24 90
7-02 (mean)
The fact that the results of these different methods are
in close agreement with each other not only indicates that
the deflexion metbod is practically equivalent to the null-
method, but also that the intensity of the magnetic field
coming into play is constant, at least in the range in which
the measuring tube displaces itself by the magnetization.
The deflexion method is, however, much simpler in
operation than the null-method, because the latter method
requires much time, and consequently many difficulties are
likely to occur during observation, such as those caused by
the convection current due to the heating of the electro-
magnet. Hence after it was ascertained that the deflexion
of Hydrogen and some other Gases. 327
method always gives a correct value, it was used throughout
the following experiments.
Â§ 8. Nitrogen.
The literature regarding the magnetic susceptibility of
nitrogen is scarcely known. Faraday * first found that the
susceptibility of nitrogen is* paramagnetic. Becquerel f
examined the magnetism of the gas, but he could not detect
it. Quincke { found a minute paramagnetic effect for
gaseous nitrogen. Pascal Â§ concluded from the study of the
susceptibility of some organic compounds containing nitrogen
that this element is diamagnetic in its gaseous state. Their
values are given in the following table : â€”
Table VII.
Date. Observer. k-.IO6.
1853 Faraday +00021
1855 Becquerel 0
1888 Quincke +0-001
1910 Pascal -00005
Thus all the previous investigators except Pascal agree
in the view that the susceptibility of gaseous nitrogen is
paramagnetic.
In the present experiment special attention was paid to
the preparation of nitrogen gas, so as to avoid contamination
with air and nitric oxide. Three different methods of
preparation were employed, and the susceptibilities of the
gases obtained by these methods wTere compared with each
other.
The first two methods consisted of the chemical preparation
of pure nitrogen gas, of which the second one was com-
paratively imperfect and served only as a check on the first,
and the last one the preparation of the so-called atmospheric
nitrogen, e. g. the nitrogen accompanying argon and other
inert gases in the atmosphere.
The method of preparation and the experimental data
obtained in the magnetic measurement of nitrogen gas thus
produced are described below in order.
The first method is the process first used by B~. Coren-
winder || in 1849. Though this is an old method, it seems
to be an excellent one for the preparation of nitrogen gas
of high purity. Lord RayleighH proved that the nitrogen
* M. Faraday, he. cit. f E. Becquerel, loc. cit.
\ G. Quincke, loc. cit.
Â§ P. Pascal, Ann. de chim. et de phys. viii. p. 1 (1910).
i| B. Corenwinder, Ami. de chim. et de phys. (8) xxvi. p. 296 (1849).
*fi Lord Eayleigh, Travers, "Study of Gases," p. 48.
328 Mr. Take Sone on the Magnetic Susceptibilities
gas thus obtained was free from any trace of the nitric oxide
which is more or less present in the nitrogen gas obtained
by most of the other methods, and the removal of which was
very difficult.
A solution of ammonium chloride and potassium nitrite
was gently heated in a flask on a water bath. After all the
air in the flask had been expelled, the gas generator was
connected with a large glass reservoir and then the gas
collected in it. Before introducing the o-as into the com-
pressing cylinder, the gas in the reservoir was passed
successively through a red-heated copper gauze, bottles
containing a solution of ferrous sulphate cooled with ice,
strong sulphuric acid, and tubes containing soda lime, calcium
chloride, phosphorus pentoxide, and calcium chloride.
The ends of the tubes and of the bottles were brought so
closely together that the rubber tubes connecting them were
exposed to the gas as little as possible, and all the connecting-
parts were covered with collodion films.
The following table contains an example of the data
obtained in' the magnetic measurement: â€”
Table
VIII.
Dec. 26, 1916.
N2
{p=25 atm.;
)
Vacuum.
2h 45m p.m.
, *=16Â°-7CL
b=20-0,
3h 25*
p.m., rf=18Â°-
0C, ft =20-0,
mAo-fâ„¢o=91-4086c
igin.
= 12 atin.)
Calculation.
4h 33m p.m.
,t=19Â°-bC,
6=200.
dNÂ»~
: -5-88+4-15= -1-73 cm..
â– ma +w0=91-3748 g
in.
d =
a
:98-00 4-4-15
= 102-15 cm.,
C.
S.
c.
mN =
=0-09278 gm.
, Â»&a=0-0590gm.
amp.
0
cm.
2-20
cm.
9800
XN2 =
-1-73K
' 0-09278
- 18-65 K,
10
0
100-20
1-60
98-60
96-90
Xa =
102-22 K _
' 00590
1732 K,
10
0
98-50
o-oo
98-50 ^ _
98-00 (meau) %a
-18-65
" 1732
-0-0108.
Three sets of such measurements gave as the specific
susceptibilities of nitrogen gas at a mean temperature 18Â° C,
10G.x^==-Â°'258> -0-272, and -0*265,
of Hydrogen and tome other Gases. 329
the mean value being
fc, = -0-26,xlO-.
The volume susceptibility of nitrogen at 20Â° and 760 mm.
pressure is
^r = -0-000309xl0-6.
The second method of preparation was due to Mai *. To
a mixture of ammonium nitrate and glycerine contained in a
flask, a few drops of strong sulphuric acid were added, and
the mixture was heated in an oil-bath at about 160Â° 0. The
generated gas was collected in a reservoir after passing it
through a strong solution of caustic potash. Before the gas
was introduced into tbe compressing cylinder, it was passed
through the trains of purifiers, which were the same as
those used in the former case, except that in this case the
solution of ferrous sulphate was replaced by a strong solution
of caustic potash.
One example of the experimental data of the measurement
for the nitrogen prepared by the second method is given in
the following table : â€”
Table IX.
Jan. 17, 1917.
N2 (p
= 25atm.)
Vacuum.
4h ;
p.m., *=19Â°-0C., b
=200,
41' 50ra p.m., *=19Â°-5 C., 6=20-0,
mN*+mo
=89-4828
gm.
mo =89-3651 gm.
C.
S.
0.
'Â».
C.
S. 8.
lV
amp.
cm.
em.
cm.
amp.
cm. cm.
cm.
0
57-70
- 9-35
-10-40
0
59-80 -6-50 -
â– 7-00
10
48-35
-11-45
-10-47
10
53-30 -7-50 -
â– 7-90
0
59-80
- 9-50
-10-38
0
60-80 -8-30 -
â€¢8-15
10
50-30
-11-25
-10-28
10
52-50 -8-00 -
-7-85
0
61-55
- 9-30
-10-32
0
60-50 -7-70 -
-7-65
10
0
52-25
63-60
-11-35
10
) o
52-80 -7-60 -
60-40 -7-60 â€”
-7-60
â€” 1037 (mean
10
52-80
-7'69(mean)
Air
(p=l atm.)
Calculation
61
1 P.M., t=
= 18Â°-5C,
6 = 20-0.
9 K
= 0^049 =1895K'
0
68-30
1-50
*m
22-76
= fs?7r= -00120
10
69-80
1-60 (mean)
/Ca
1895
* J. Mai, Ber. d. DeuUch. Chem. Ges. iii. p. 3805 (1901).
330 Mr. Take Sone on the Magnetic Susceptibilities
Two sets of such measurements were made for the nitrogen
obtained by the second method at a mean temperature 18Â° 0. ;
the results are
106.%^=- 0'288 and -0-287,
the mean value being
Xlf = -0-288 xlO-6.
The volume susceptibility at 20Â° C. and 760 mm. pressure is
^=_0-000336xl0-c.
Thus the value of the susceptibility of nitrogen obtained
by the second method is about 8 per cent, more diamagnetic
than that obtained by the first method, but it was sufficient
as a check and no further study was made of the cause of
the deviation.
In the third method the atmospheric air was introduced
into the reservoir throuo-h the bottles, containing a solution
of caustic potash, calcium chloride, concentrated ammonia
solution, and red-hot copper gauze. Before filling the gas
in the compressing cylinder it was passed through bottles
containing strong sulphuric acid and calcium chloride, and
then through the trains of purifiers, which were exactly the
same as those used in the preceding experiment.
The following are the data in the determination of the
susceptibility of the atmospheric nitrogen : â€”
Table X.
Jan. 11,
1917.
N
'' (jp=30 atm.)
Y ACTUM.
6* 10m p.m., *r=15Â°-0 C, 6=20-0.
8h 40m p.m., t=
= 14Â°-7C,
6=20-0,
mN2
+ mo= 89-4916 gm.
â„¢0=89-351Sgm,
C.
S.
d.
C.
S.
0.
. 'Â»â–
amp.
cm.
cm.
amp.
cm.
cm.
cm.
0
60-00
-11-80
0
5600
-6-50
-8-00
10
48-20
-11-80
10
49-50
-9-50
-8-35
0
60-00
-12-50
0
59-00
-7-20
-7-70
10
47-50
-11-70
10
51-80
-8-20
-8-10
0
59-20
-11-40
0
60-00
-8-00
-8-35
10
47-80
- 10-80
10
52-00
-8-70
-860
0
58-60
-12-10
0
60-70
-8-50
-8-75
10
46-50
-11-80
10
52-20
-9-00
-8-55
0
58-30
â€” 11 -74 (mean)
0
61-20
-8-10
-7-65
10
5310
-7-20
-7-65
0
60-30
-8-10
-795
10
52-20
-7-80
0
60-00
-8-15(1
of Hydrogen and some other Gases. 331
Aik (p=l atrn.).
Calculation.
10h 5m
?.M
., <*=12Â°-0C
!., 6=20-0,
**â– ,=
:- 11-74+8-15= -3-59 cm.,
m
,+*Â».:
gm.
C. '
S.
d.
*a=
=0-27+8-15:
=8-42 cm.,
amp.
CUl.
cm.
mw
"o.2 Poo Po2
where Xa-> <%0 , ^ , and %A are the specific susceptibilities of
air, oxygen, nitrogen, and argon, and p0 , pN , and p4 are
the ratios of the masses of these gases in air to the total
mass of air respectively.
Then the value of the susceptibility of argon can be
deduced in the following way, taking the percentages of
nitrogen and argon in air as follows : â€”
75*5 and 1*3 per cent.
Hence the percentage of argon in atmospheric nitrogen is
1-3x100
75-5 + 1-3
1*7 per cent,
But the presence of argon in the atmospheric nitrogen
produces an increase of the dia magnetic susceptibility from
Xk= -0-265 xlO"6 to Xw .,= -0-360x10-Â°.
Hence the susceptibility of argon will be
= ~-0-360xlQ-G _ 0-983 x (-0-265) X3Q-6
Xa~ 0-017 ' 0017
= -5-86x10-Â°.
Taking as the weight percentage of oxygen in air 23*15
and introducing the values for %#, %^ , and %A in the equation
for oxygen, we get
= 23-85 x 10^6 _ 0-755 x (-0-265) x 1Q-3
Xo~ 072315 0-2315
0-13 x (-5-86) xlO"6
0-2315
= (103-0 + 0-9 + 0-3) xlO"6.
= 104-2 x 10"c.
But the susceptibility of oxygen directly determined being
%O3 = 104-,xl0-Â«,
these two are in fair agreement with each other.
of Hydrogen and some other Gases. 333
It becomes now clear that the volume susceptibilities of
the nitrogen and argon present in the atmosphere are not
negligibly small, as was believed to be the case ; and that
the sum of their values amounts to a little above one per
cent. of the susceptibility of air.
Â§ 9. Hydrogen.
Hydrogen is one of the gases the magnetic susceptibility
of which has been comparatively well studied. Nevertheless,
owing to the difficulties which accompany the experiments,
we have as yet no reliable experimental value of its magnetic
susceptibility; even its sign was not decided till quite
recently. But recent investigators seem to agree in the
view that hydrogen has a diamagnetic susceptibility, and
now the determination of an exact value of its susceptibilitv
becomes an outstanding problem.
Quincke * first determined the magnetic susceptibility of
hj'drogen, and found as its susceptibility a value
K= +0-0003 xlO"6.
Bernstein f published the results of his experiments on the
magnetism of some gases in his dissertation at Halle, and
gave as the susceptibility of gaseous hydrogen,
k= -0-005 xlO"6.
Assuming the additive law, Pascal % calculated the atomic
susceptibility of hydrogen from the study of some organic
compounds ; the result of his calculation being
x= -3-05x10-Â°.
Kammerlingh Onnes and PerrierÂ§ measured the magnetic
susceptibility of liquid hydrogen and found as its volume
susceptibilitv,
k= -0-186 xlO"6.
Taking for the density of liquid hydrogen a value
obtained by Dewar 0'07, he deduced as the specific
susceptibility of liquid hydrogen,
x= -2-7 xlO"6.
No account of the experimental details was given in their
* G. Quincke, loc. cit. f Bernstein, loc. cit.
X P. Pascal, loc. cit.
$ H. KammerliDgli Onnes and A. Perrier, Amsterdam Pruc. xiv.
p. 121 (1911).
334 Mr. Take Sone on the Magnetic Susceptibilities
paper, but the authors did not claim too much weight for
their result and were satisfied with the fact that their result
roughly agreed with the value obtained by Pascal.
More recently Biggs* carried out the same investigation
of gaseous hydrogen. He tried to overcome the experimental
difficulties by utilizing the large absorptive power of
palladium for hydrogen, and found too large a value for the
susceptibility of the gas.
The chief" difficulties met with in the determination of
gaseous hydrogen are two : the first is the preparation of
pure hydrogen free from oxygen, and the second the deter-
mination of the magnetic force exerted upon the gas, owing
to the smallness of its volume susceptibility and density.
In the present experiment these two difficulties were over-
come, and a highly trustworthy value for the susceptibility
of hydrogen gas was obtained. The details for the prepar-
ation of the pure gas and the determination of its magnetic
susceptibility are given in the following pages.
(a) Preparation of pure Hydrogen Gas.
In the present investigation the pure hydrogen was
prepared by the method which Morley | used in the deter-
mination of the volumetric ratio of hydrogen and oxygen in
water. The chief differences between his and the present
case were in the construction of the decomposing cell and
the combustion tube. In my case an open vessel was used
as a decomposing vessel, and platinum electrodes were
introduced into the cell by insulating the leading wires with
glass tubes. Oxygen which was generated at the anode,
was allowed to escape into the atmosphere. The electrodes
were separated from each other as far as possible in order to
lessen the diffusion of oxygen from the anode to the cathode.
The open end of the vessel was covered with a mica plate ;
besides, the upper portion beino;' wholly wrapped in cloth in
order to prevent any contamination of liquid.
In the early period of the present experiment, I used for
a long time a combustion tube of hard j>lass, which contained
pieces of reduced copper gauze, both ends of the tube being-
ground and joined to the rest of the purifying train. But
in the course of the experiment it was found that in a long-
period of time the joints became gradually loosened by
repeated heating and cooling, allowing the diffusion of air
* H. F. Bigga, Phil. Mag. xxxii. p. 131 (1916).
f E. W. Morley, Amer. Journ. Sci. iii. xli. p. 220 (1891).
of Hydrogen and some other Gases. 335
into the tube : hence finally I used palladium asbestos
instead of copper ; and in this case the heating temperature
being below 250Â° C, an ordinary glass was used as the
combustion tube, both ends of the tube being fused together
to the rest of the apparatus. In consequence of this change
of arrangement, not only was the measurement of suscept-
ibility of hydrogen greatly facilitated, but the results of the
experiment became quite consistent.
The hydrogen generator, purifying train, the reservoir,
and the compressing cylinder, together with the measuring
tube, are shown in fig. 8. a is a large cylinder having a
Fijr. 8.
capacity of 4 litres. About 3 litres of pure dilute sulphuric
acid was put into it ; the concentration of the solution wras
the same as used by Morley, e. g. it contained one-sixth of
pure sulphuric acid in volume, b is a glass tube placed
concentrically with the cylinder in a. The anode is placed
in the upper portion of the cylinder, while the cathode is
placed at the lower part of tube b. Each electrode has an
area of 25 sq. cm. C is a bulb for preventing the passage
of the acid fumes upwards, d is a large glass bottle filled
336 Mr. Take Sone on the Magnetic Susceptibilities
halfway with a 50 per cent, solution of caustic potash and
placed obliquely, e is a long horizontal tube containing fine
pieces of glass wetted with the above solution, /is a bulb
for preventing the mixing of the solution of caustic potash
with sulphuric acid in a vessel g ; h is a soft glass tube
loosely filled with palladium asbestos, and i a thermometer
graduated to 360Â° C. j is a wash-bottle containing strong
sulphuric acid, k is a cock dipped in a mercury bath,
which prevents the diffusion of hydrogen to the outside and
that of air to the inside. I is a tube containing molecular
silver, which serves as a sensitive indicator of hydrogen
sulphide, m is a large cylindrical tube in which rods of
caustic potash are placed, and n a cylinder containing
phosphorus pentoxide. o is a large glass bulb having a
capacity of about 3 litres, p and q are two glass cocks, the
latter being attached to a side tube leading to a Gaede
auxiliary pump. All the stopcocks described above are
lubricated with Yakuum-Hahnfett and dipped in small
mercury baths. The end of the delivery tube r is brought
in contact with the tapered end of the compressing cylinder s
at v and covered with a thick rubber tube, and this connected
portion is dipped in a mercury bath u. The portion extending
from the generator to the end of the delivery tube is wholl}'
made of glass having no ground joint, except the portion of
the stop-cocks, and is therefore entirely safe from the leakage
of gases for several months.
As the rubber connexion between the compressing cylinder
and the delivery tube was used only when the compressing
cylinder was to be filled with the gas, the diffusion of air
through the rubber tube was negligible ; and the gas in the
compressing cylinder was proved by an experiment â€” which
will be described later â€” to be perfectly free from air, so
that I could safely use this connexion throughout the whole
experiment.
(b) Filling the Measuring Tube with the Gas.
The method of filling the compressing cylinder with
hydrogen is as follows : â€”
To drive off the air contained in the purifying train and
the reservoir, a current of hydrogen generated in the
decomposing cell, at the rate before described, is passed for
about two days, while the palladium asbestos is heated to
200Â° 0. After almost all the air has been driven out of the
of Hydrogen and some other Gases. 337
apparatus with the hydrogen, the stop-coek k is closed, and
the gas contained in the bulbs m, n, and o pumped out by the
Graede auxiliary pump, and then the cock q is closed and h
opened to introduce the hydrogen into the bulbs m, n, and o.
When they are filled with gas the evacuation is a^ain made,
and then the new gas introduced. After the same process
has been repeated several times the cock p is finally closed
and the reservoirs are filled with gas till the pressure of
the gas reaches about 2 cm. higher than the atmospheric
pressure. This pressure is attained when the surface of water
in the cylinder b of the decomposing cell reaches the lower
position, as shown in fig. 8.
To fill the measuring tube and the compressing cylinder
with gas, the tapered end of the latter is brought in contact
with the end of the delivery tube, and then the air in them
is pumped out. When the cock q is closed and the cock p
opened, the gas stored in the reservoir rushes into the
evacuated cylinder, filling it almost instantaneously at a
pressure of about one atmosphere. Next the cock k is
opened and new gas supplied to the reservoir, till the former
pressure is again attained in about 15 minutes. Thus the
cylinder and the measuring tube are always filled with
hydrogen at about one atmospheric pressure ; they are in an
evacuated condition only for a moment, when the communi-
cation to the pump is stopped by the cock q and the cock p
opened to introduce the gas into the cylinder. Thus any
diffusion of air from outside through the packings and
connexions is completely prevented. The same process of
washing the interior of the measuring tube and the com-
pressing cylinder with pure hydrogen is usually repeated
live times and the gas finally introduced into the tube and
the cylinder is employed for the measurement.
The method of measurement of the mass of the hydrogen
was to determine the mass by replacing water in a eudiometer
with the hydrogen stored up at high pressure in the
measuring tube. If the pressure and 'temperature at the
time of measurement of the volume are known, then by
taking the vapour pressure at that temperature into con-
sideration, we can calculate the volume of gas at standard
conditions. Multiplying the density at 0Â° C. into the volume
obtained above we obtain the mass of the gas. In some
cases the determination of the mass of hydrogen was also
made by weighing and comparing the weight with the results
of the volumetric measurement.
Phil. Mag. S. 6. Vol. 39. No. 231. March 1920. Z
>38 Mr. Take Sone on the Magnetic Susceptibilities
(c) Results of Magnetic Measurement.
About thirty independent observations were made for the
pare hydrogen gas ; the following table contains the data
for the magnetic susceptibility of the gas obtained in twelve
sets of measurement which give the most reliable results.
In the table, vnH* ts: , %H ,ma, ta, and %a denote the masses,
temperatures, and specific susceptibilities of hydrogen and
air respectively. The experiments are arranged in a
chronological order.
Table XI.
No.
= 4'21 x : 1016/sec,
and therefore
o-0=l-117 xlO1, E.M.U.
Hence, at a given temperature, the value of susceptibility
71
depends on ~KX202. In order that the above expression
may give the observed value, we must take
|Kf202 = 393-5xl010ergs,
or Oo = 6-54xl014/sec.
This angular velocity corresponds to the frequency
y=l-04xl014/sec.
of the infra-red radiation. A similar expression holds also
for the susceptibility of other diatomic gases.
According to Bohr, the helium atom has two positive
nuclei at the centre of the circular Orbit, in which two
electrons are revolving with a constant velocity. If we
assume that the helium atom has a similar structure to the
hydrogen molecule, in which two positive nuclei are situated
very near to each other, the atom may possess a similar
characteristic rotation as in the case of the hydrogen
molecule. We have then
. x~ 4?iKry
According to Bohr,
r = 0-318xl0-8cm., a> = 19 x lO^/sec.
Therefore
o-0=l-81 xl03E.M.U.
In order therefore that the above expression may give a
value of susceptibility %= â€”11*0 X 10~6, as actually observed
by Tanzler |, wre must take
^KO02=7-84xl010ergs,
or O0 = 3-80xl015/sec.
* N. Bohr, loc. cit. : P. Debye, loc. cit.
t P. Tanzler, loc. cit. The present value is calculated from the
original value by taking v =104TxlO-Â°,
of Hydrogen and some other Gases. 319
But since X20 is here not very small as compared with g>,
we cannot use Bohr's values for co and r, and consequently
no will acquire a somewhat different value from that above
given .
Thus Bohr's models for hydrogen and helium atoms give
by magnetization a diamagnetic polarization, whose values
are of a right order of magnitude, provided that there is a
definite rotation about an axis perpendicular to the line
joining two positive nuclei, the velocity of which is very
small compared with the velocity of revolving electrons.
According to the above results, monatomic and diatomic
gases are all diamagnetic, and their susceptibilities can be
calculated, provided the models of the molecules are known ;
this conclusion generally agrees with the observed facts.
One important exception is the case of oxygen, which is a
diatomic gas. Bat if we assume that in virtue of the form
and nature of oxygen molecules, the characteristic rotation
is completely absent and the rotational energy of revolving
electrons is relatively small, then in place of the expression
for susceptibility above given, we have *
% =
3
gxiW + KT)'
where &>0 is the angular velocity of revolving electrons,
K the moment of inertia of the electron about its axis, and
RT the rotational kinetic energy corresponding to Langevin's
paramagnetism. Hence % is positive and varies hyper-
bolically with temperature. Thus by assuming particular
conditions, the paramagnetism of oxygen gas may be
explained.
In the case of polyatomic gases, the resultant magnetic
axis of the molecules is not generally perpendicular to the
axis of the characteristic rotation, as in the case of the
diatomic gases. In this case, if we resolve the magnetic
moment into two components parallel and perpendicular to
the axis of the characteristic rotation, the former produces a
paramagnetic effect and the latter a diamagnetic ; the
observed polarization is the sum of these two effects, which
may be positive or negative. Thus the polyatomic gases are
paramagnetic or diamagnetic according to their natures.
^ In the above discussion the molecules are assumed to be
rigid and no account is taken of the Langevin diamagnetism.
If the small term of this diamagnetism be introduced, the
above conclusions do not materially change.
* K. Honda and J. Gkubo, he. cit.
350 Mr. Q. Cook : An Experimental Determination
Thus the magnetization of different gases can be satis-
factorily explained by the theory of Professors K. Honda
and J. Okubo by introducing certain assumptions, not only
qualitatively but also quantitatively.
In conclusion I wish to express my deepest obligation to
Professor K. Honda, under whose suggestion and constant
guidance the present investigation was carried out. I also
wish to express my hearty thanks to Professors M. Ogawa
and M. Katayama, of the Department of Chemistry, for
their valuable suggestions and criticisms in the work. My
thanks are also due to Messrs. N. Yamada and K. Shikata
who kindly helped me in the preparation of pure gases in
the early part of the present research.
Physical Institute
of the University, Sendai,
April 26, 1919.
XXX. An Experimental Determination of the Inertia of a
Sphere moving in a Fluid. By Gilbert Cook, Af.Sc,
A.MJnst.C.E*
IT has been shown by Stokes t that when a solid body is in
motion in a frictionless fluid of infinite extent, the
effect of the fluid pressure is equivalent to an increase in the
inertia of the body. The effect is manifest only in cases of
accelerated or retarded motion, and has a direct application
to the dynamical theory of the oscillating mine, which the
writer had occasion to study during the war.
The only cases for which the magnitude of the increase of
inertia hns been determined analytically are those of the
cylinder of infinite length moving at right angles to its axis,
and of the ellipsoid of revolution, including the sphere. The
case of the sphere, in which the increase of inertia is found
to be one-half of the mass of fluid displaced, is one which
lends itself readily to experimental verification.
The experiment was carried out in a tank 15 feet in
diameter and 30 feet in depth. The spherical body consisted
of a mine-case 38*2 ins. in diameter ballasted in such a
manner that its weight in water was approximately one
pound, the displacement being 1080 lb. ' It was allowed to
fall freely through the water under the influence of gravity.
The motion was recorded by means of an instrument designed
* Communicated by Prof. J. E. Petavel, F.R.S.
f Trans. Camb. Phil. Soc, vol. viii. (1843) ; ' Collected Papers," vol. i.
p. 17.
of the Inertia of a Sphere moving in a Fluid. 351
by the writer for the study of oscillatory motions under
water, a light cord attached to the mine-case passing round
the drum of the instrument, and equal intervals of time being
indicated by a small lateral movement given to the recording-
pen electrically from a clock. From the record thus obtained
the velocity and acceleration at any instant could be measured.
The friction of the instrument was negligible, but a counter-
poise applied for the purpose of maintaining a small tension
in the cord reduced the downward force on the mine-case bv
0-OG lb.
The equation of motion may be written : â€”
at 7
where M' is the inertia of the body, v the velocity, F the
vertical force due to gravity (corrected for counterpoise of
the recording instrument), and kvn a term expressing the
frictional resistance to the motion of the body through the
fluid.
The correct value of n will be such as to make the relation
between -- and vn linear. As the frictional resistance is
at
mainly that due to skin friction and eddy formation, it might
be anticipated that n wTould have a value approximating to 2.
The values of the velocity, acceleration, and square of
velocity are given in the following table :â–
VJ **AV fe*
o """^^ â€¢
dv
Time
v.
di
(sees.).
(ft. sees.).
(ft. sees.2).
v2.
2-5
0-1333
0-01814
0-0178
5
0-1768
0-01667
00312
7-5
0-2195
0-01745
0-0482
10
0-2631
0-01747
0-0691
12-5
0-3026
0-01412
00916
15
0-3383
0-01440
0-1145
17-5
0-3711
0-01187
01378
20
0-3988
0-01024
0-1592
22-5
0-4233
0-00936
0-1792
25
0-4458
0-00867
01990
27-5
0-4683
0-00932
0-2193
30
0-4895
0-00760
0-2395
32-5
0-5075
0-00680
0-2575
35
0-5238
0-00627
0-2748
The relation between -=- and v2 is plotted in fig. 1. It will
be seen that the plotted points lie close to the straight line
y= 0-0193 -0-0494 ^.
352 Determination of Inertia of a Sphere moving in a Fluid.
The vertical force due to gravity was 0*950 lb., so that by
comparison with the equation
,dv
M
dt
F-h
the value of M' will be seen to be 1584c lb., or 1*46 times the
displacement of the bodv.
1%. 1.
OP..O
oiS
o x
0
o
x\
OIO
<3
0s
O ^v
O ^
c
V o
005
^
X.
\
\
.
(Velocity)^
O 'OS -to -15 '20 '25 '30
The value of k, the coefficient of resistance, expressed in
gravitational units, is 2*44: for this particular case. Assuming
the resistance R to be proportional to the square of the
diameter, so that
n=k'd2v2,
the value of k' is 0'241, the units being feet, pounds, and
seconds.
Having in view the modification of the theoretical flow
which may be caused by frictional resistances, the value
obiained for the increase' of inertia, namely 0'46 times the
displacement, may be considered a sufficient verification of
the value deduced analytically for this case. It may be
pointed out that any error involved in the assumption of the
value 2 for the index of n, whilst modifying the frictional
coefficient, would affect the value of the inertia obtained from
the experiment by a negligible amount. Although the fluid
was not of unlimited extent, the ratio of the dimensions of
the tank to those of the sphere were sufficiently large to
render the effect of the cylindrical walls inappreciable.
t 353 ]
XXXI. A New Cadmium Vapour Arc Lain]?.
By Frederick Bates *.
r|1HE necessity for increasing the intensity as well as the
-I number of monochromatic light sources has frequently
been emphasized during the past few years. Unfortunately
but little has been accomplished toward attaining this
objective. In 1906, the writer f directed attention to the
importance of this subject and suggested that the so-called
yellow-green line (X = 5461 A) of incandescent mercury
vapour be adopted as the source for standardization purposes
in polarimetric work. The quartz-mercury vapour-lamp
was a great advance in that it provided not only the yellow-
green line, -but several additional lines of lesser intensity.
The best available methods of optical purification are such
that a monochromatic source is of little A^alue unless the line
is sufficiently removed from its immediate neighbours that
nearly complete separation by spectrum filtration is possible.
If sufficient light to satisfy modern practical and research
needs is to be obtained from any such source it is necessary
to use a relatively wide slit, with a consequent probable
inclusion of other wave-lengths in the immediate vicinity of
the one desired.
When the most intense ol all known light sources,
namely, the direct radiation of the sun, is utilized, the
necessary slit width, while less than that for any other
known source, must still be such as to include a relatively
large number of wave-lengths. The resultant wave-length
or so-called optical centre of gravity of such a group of
waves can be considered as a monochromatic light source in
only a very restricted sense, and finds effective application
in but few fields of work. It is especially unsuited to the
^tudy of phenomena which change rapidly with change of
wave-length. The necessity for obtaining additional intense
light sources is consequently imperative.
Among the possible sources which have been suggested is
that of the rotating arc with cadmium-silver alloy electrodes.
This source gives a number of fairly intense lines sufficiently
isolated from each other and fairly well distributed through-
out the spectrum. The writer has carried out many experi-
ments with tliis source, using an improved rotating arc. It
* Communicated by the Author.
t Bulletin of the Bureau of Standards, vol. ii. p. 2-'S(.>.
Phil. Mai). S. 6. Vol. 3D. No. 281. March 1920. 2 A
354 Mr. F. Bates on a New
was found impossible to maintain an arc sufficiently free
from flicker to give satisfactory results.
Another possible source experimented with is the quartz
cadmium vapour arc lamp, described by Lowry and Abram *.
This lamp is always unsatisfactory owing to two defects.
It is necessary to have it permanently connected to an air-
pump and to immerse the electrodes in water. If the
cadmium in a vapour lamp is sufficiently pure, the adhesion
between the cadmium and the quartz results in the de-
struction of the lamp upon the solidification of the cadmium.
An improved form of lamp has been brought out by Sand |.
In this type, the tendency of the cadmium to adhere to the
quartz walls is stated to be lessened by introducing into
the lamp a small amount of zirconia in the form of tine
powder. The cadmium is placed in a side tube connected
to the pump and the body of the lamp by a tube constricted
to three capillaries for the purpose of filtering the metal.
Additional filtering may be obtained by introducing a roll
of iron gauze. Extensive experiments by the writer with
this type of lamp have demonstrated that it is impracticable,
provided a pure cadmium spectrum is desired. The method
of filtering suggested is inadequate. The impurities intro-
duced into the lamp by this method of tilling undoubtedly
have, a tendency to prevent breakage, but effectively prevent
obtaining a relatively pure, intense cadmium spectrum. In
order to eliminate all oxide and other impurities from the
cadmium used in filling, it is necessary to carefully distil
the cadmium into the body of the lamp. Upon allowing the
lamp to cool, adhesion between the quartz and the metal
takes place in spite of the presence of the zirconia. If the
lain}) does not crack upon the first solidification of the
cadmium, thin sections of the quartz are peeled from
the walls by the contracting metal. Upon cooling a second
time, the lamp was invariably cracked.
Numerous experiments of varied character failed to over-
come the constant breakage of the Sand lamp. Among the
filling mixtures tried was a cadmium-mercury alloy. The
percentages of the constituents were varied on a wide range.
The introduction of the mercury is very effective in pre-
venting the cracking of the lamp, as the alloy formed was so
soft that no appreciable adhesion between it and the quartz
resulted. It was found, however, impossible to obtain
a brilliant cadmium spectrum under any circumstances.
* Trans. Faraday Soc. vol. x. p. 103 (1914).
t Proe. Phys. Soc. London, vol. xxviii. p. 94 (1915-16).
Cadmium Vapour Arc Lamp. 355
The vapour-pressure of the mercury being so much higher
than that of the cadmium, resulted in the electric energy
being almost entirely carried by the mercury, and the usual
brilliant mercury spectrum resulted.
In view of the preceding facts, it is evident that a
serviceable brilliant cadmium-vapour lamp might be ob-
tained by alloying the cadmium with a suitable element of
lower vapour-pressure. Through the courtesy of Dr. W. F.
Hillebrand, a quantity of the little-known element, gallium,
was obtained. The material was in a very impure con-
dition, containing approximately 10 per cent, indium. The
freezing-point was below 22Â° C, at which temperature it was
a liquid with a viscosity less than that of mercury. A study
of the impure material was made by Dr. G. E. F. Lundell,
who succeeded in obtaining the gallium in a relatively pure
condition.
Crude gallium was dissolved in aqua regia, treated with
sulphuric acid and famed to remove nitric acid. After
dilution, small amounts of lead sulphate were filtered off.
The solution was then diluted, treated with hydrogen
sulphide and filtered to remove the hydrogen sulphide group
of elements. The filtrate was boiled to expel hydrogen
sulphide and treated with ammonium hydroxide. The
precipitate was filtered oft', dissolved, and reprecipitated
three times to free it from zinc. The final separation from
indium was based on the solubility of gallium hydroxide in
a solution of sodium hydroxide and the insolubility of indium
hydroxide in that reagent. The sodium hydroxide separa-
tion was carried through three times. The deposition of
gallium was finally carried out by electrolysis of the alkaline
solution as recommended by Uhler and Browning*.
The purified gallium had a freezing-point of approximately
30Â° 0. This surprising fact has since been verified by the
careful work of Richards f, who has definitely fixed this
temperature at 30Â°' 8 0. Regarding the boiling-point of
this element, but little is known. The few experiments
which have been made are in agreement that it is above
1500 degrees C. This property should make it an ideal sub-
stance for the purpose in hand, provided it would alloy with
cadmium. The first experiment demonstrated that it united
with the cadmium with the utmost ease. In fact, the
addition of a few drops to ten or fifteen cubic centimetres
of cadmium completely changed the texture of the latter,
* Am. .lour. Sci. cxcii. ixlii. Fourth Series) p. 389 (1916).
t J Miir. Am. Oliein. !Soc. vol. xli. p. 181.
2 A 2
356
Mr. F. Bates on a New
rendering it relatively soft and greatly reducing its tensile
strength. Subsequently it was discovered that upon dis-
tilling the cadmium from the alloy at a pressure of
O'OOI mm. of mercury, the minute quantity of gallium
carried through was sufficient to completely change the
character of the cadmium and to prevent adhesion between
the cadmium and the walls of the lamp.
Fig.l
The type of quartz lamp used in the experiments is that
shown in figure 1. The total volume is approximately
ten cubic centimetres. The electrodes consist of tungsten
wires (B) entering through quartz capillaries. They are
closed with lead seals similar to the type described by
Sand *.
* Proc. Phys. Soc. London, vol. xxvi. p. 127 (1914).
Fijy. 2.
Cadmium Vapour Arc Lamp, 357
In filling the lamp, the cadmium containing
two or three per cent, gallium is placed in the
bulb F. It is necessary to maintain the pressure
in the lamp and connexions below 0*001 mm. of
mercury with the exception of that due to the
cadmium and gallium, throughout the process of
distilling. Owing to the fact that the volume
of the lamp is relatively small, the quartz capil-
lary at E should be of such a length as to permit
of sealing off in the shortest possible time. The
flame used for this purpose should be small, and
the heating of the tube on both sides of the
capillary should be prevented as far as possible.
The method indicated above, if carefully
followed, will give a lamp with indefinite life.
One of this type has been in intermittent use for
over a year and shows no sign of deterioration.
Should traces of oxide or stains due thereto
appear during the process of rilling, they can
readily be reduced by introducing pure dry
hydrogen and heating. The lamp may be started
by heating with a flame to vaporize the metal.
It is in all cases advisable io have a current of
air blowing upon the lead seals to keep them
cool. If the blast is allowed to strike the body
of the lamp, the cadmium is condensed and
obscures the arc. The most convenient source
of energy for operation is the ordinary 110 volt
lighting circuit, on which it will operate con-
tinuously with a current as small as 3 am]).
and a drop of 14 volts across the terminals of
the lamp. The most satisfactory results, how-
ever, are secured with a current of about 7 amp.
and a drop across the terminals of about 25 volts.
Under this condition a practically pure cadmium
spectrum of great brilliancy is obtained. The
intensity secured is apparently equal to that
which would be obtained were the lamp filled
with cadmium alone. The map * of the spec-
trum of gallium given in figure 2 is interesting.
The wave-lengths and intensities of the lines
are given in Table I. in this connexion. It
will be observed that there are but five lines in
the visible spectrum and that from practically
4200 A to 6400 A there are no lines.
* Eder & Yalenta, Atlas Tyjji with GH.
Taking D as the origin let DF = ?< and DG = .i'.
* Communicated bv Prof, G. W. Todd, D.Sc,
360
Mr. S. P. Owen on Radiation
If N = normal radiation from the element at G, the total
radiation received from G at F
N cos 0 cos sin <Â£ . r . dr dd dx dc
d?
where r .drd6 = sive& of the element at F,
dxdc â€”area of the element at G ; dc being an ele-
ment of the circumference of the cylinder,
and d = FG.
This will be absorbed by the disk if the latter be a perfect
absorber of radiation.
The total radiation received by the whole plate from the
whole cylinder
= 2 f 2na P f 8m " f DB Ncos^cosÂ»:siu^Â»yi
Jo J*x Jo Jl)A <1-
where xx~ DK and #g = DM,
r/r d# . dx . dc
, . (1)
from a Cylindrical Wall. 361
In II., taking OAB = Â«,
DA = a cos 0â€” b cos a,
DB=acos0 + />eosa,
and &sirfa=asin #.
DA = ^ cos 0- v/^^in7^
and DB = rÂ« cos (9 + vA7"-^2 sin" 6.
Jit I., cos <Â£> = - = /=_^=â€” sin (i â€” â€” ^== = .
' // / "J i 2 ' ' / 2 i 2
" V â€¢' + i" v ts +â– r*
Hence (1) becomes
2 f ^ c** rin Â« rDJ' ncosa..^2^.^.^.^
Jo J, Jo Jda/ (tf2 + r2)2
Integrating with respect to c and then to ?' we get
rCx* 7 fsin _l7' cos Or .DB TDA
I 1 #tf#l fan-1 â€” tan
J.r, Jo loc L a? a?
J,, Jo l'# L
-isin2tan-^ + isin2(tan-^)"
r/0.
Using the above values for DA and DB we get after
reduction
, _XDB ,DA 2tf\A2-a2sin20
tan 1 â€” -tan"1 â€” = tan , 2 , , M â€” :
therefore as part of the integral with respect to 0 we get
_iÂ»
cos 0 tan v- 0 ,/0
Jo ~~`cos(9 _ rf
~ Jo vA2â€” a2sin20 f 1 + A2(62-.a2 sin2 0) }
0 by integration by parts,
ere A = . â€” - ,
^2 + a2â€” h'
3^2 Mr. S. P. Owen on Radiation
Putting Â«sin# = Â£sin<Â£, this reduces to
IT
pA52sin2^ 6cos^.^j>
= Aft2p sin2. ^
. V â€¢ (2)
b
(* sin-1 a T)p>
Taking the part I cos 6 . sin 2 . tan"1 â€” cW,
Jo ^;
, b
?. 2 â€” ft2 sin2 0=//,
the integral reduces to
Similarly I cos 6 sin 2 . tan"1 .2 + 2a/2)2-4^
Â§ 2. Experimental Verification .
Using the above result, the radiation constant o- was
determined by the following experiment :
The apparatus consisted of an ordinary steam jacket, the
inside of which was covered by an even layer of lampblack
obtained by the burning of camphor. This was fixed in a
vertical position.
The amount of heat radiated to a copper plate .was
measured by means of a modification of Bunsen's Ice
Calorimeter. The plate consisted of the bottom of a copper
calorimeter, enclosed in a block of wax, the bottom being
exposed, sheer with the surface of the wax. A capillary
tube, previously calibrated, was fitted through a rubber
stopper to the top of the calorimeter. The inside of the
vessel was filled with crushed ice and water, and by fitting
in the stopper the capillary tube could easily be filled with
the water from the vessel. By observing the rate at which
the meniscus in the capillary moved, the amount of contrac-
tion and hence the amount of heat given up to the ice in
unit time could be calculated.
from a Cylindrical Wall. 365
In order to eliminate the heat radiated from the sur-
roundings, a guard-ring filled with crushed ice was used.
This was made so that the plate could be completely cut off
from exposure to the cylinder, and by means of an aperture,
equal in size to that of the plate, it could be exposed to the
cylinder at will.
Observations were taken at half-minute intervals, a few
minutes with the plate covered, then exposed, and finally
with the plate covered again. The mean of the first and
last set of values was subtracted from the mean of the
second set and thus the heat radiated from the cylinder was
obtained.
In the experiments, which were conducted with two
cylinders of different heights, the diameter of the plate was
practically equal to that of the cylinder, thus putting in the
expression (5) a = b, it simplifies to :
11
= a (IV - T24) \ { V** + ^ ~ *2 } "â€¢
c.c.
This expression was used in the calculation :
Average radius of plate and cylinder = 2'35 cm.
Volume of capillary per cm. length =r â€¢0201
Mean contraction per minute when plate was
covered = C] cm.
Mean contraction per minute when plate was
exposed = c2 cm.
IV %*â€¢ xi- -lV ci- c2- o--c.,. irXlO5.
373Â° ubs. '273Â° abs. 1*1 cm. 23 '3 cm. 2-3 cm. 2'88cm. -58 cm. 4*88
l"5 â€ž 21-8 â€ž 1-26,, 1-76 â€ž -5 â€ž 4-88
2'1 â€ž 24-3 â€ž -73., 112 ., 39 â€ž 4"94
The accepted value of o-=5*32 x 10"5 ergs, cm.2 sec. de<
In conclusion. I must express my great indebtedness to
Dr. (t. W. Todd, who suggested the problem and whose
advice and criticism lias been invaluable in the experimental
work.
Royal Grammar School,
Newcastle-on-Tynej
Sept. 1919.
t 366 ]
XXXIII. On the Measurement of Time â€” a Rejoinder to
Dr. N. Campbell. By L. Silberstein, Ph.D., Lecturer
in Math. Physics at the University of Rome *.
Tf 1HE paper on " A Time-Scale, etc.'' by the present
JL writer, published in the September issue of this
Magazine, opens the investigation by a statement that the
principle of common time-scales amounts to this : â€” -A certain
kind of motion (translatory or rotatory) is declared to be a
uniform motion ; the path is then cut up by means of
compasses etc. into a series of equal segments (or angles) and
the instants of passage of the mobile through the divisions
of this metrical scale are taken as Â£ = 0, 1, 2, 3, and so on.
Dr. N. Campbell, in the November issue of the Phil. Mag.
(pp. 652-4), believes "this statement to be untrue. " It will
be my duty to show that it is true. In the second place,
Dr. Campbell believes "still more untrue" (as if truth were
liable of gradations) " the statement implied, that time-
measurement is impossible except by some such artificial and
elaborate method as he [Silberstein] proposes/' Now,
concerning this second point, I have not said nor meant to
imply that other methods independent of space-measurement
were impossible. I simply proposed one, without excluding
the possibility of other methods being invented by others.
Thus I have nothing more to say about this second point.
A third point, however, is that Dr. Campbell offers us his
own views on the measurement of time, and these are so
palpably unsatisfactory as to require but a few words to
be refuted. :
But let me first attend to the first point. Now, my state-
ment, quoted at the outset, is not only logically true (that
is to say, that a theory of chronometry based on " uniform "
motion and paths or angles carved up into " equal " parts
would be a possible logical theory), but also, which is
of great importance, historically true, the two principles,
uniformity and rigid subdivision or transfer, being the
dominant and basal features of every practical chronometry
since times immemorial and up to, and including, our own
days. In fact, the most ancient measurement of time, as
practised by Babylonians, Assyrians, Egyptians, and whom
not, was based on the assumption of uniformity of rotation
of the heavenly sphere round the Earth, and on a rigid,
metrical subdivision of the angles involved in this pheno-
menon. Massive columns were erected and carefully kept
* Communicated by the Author,
On the Measurement of Time. 367
for this purpose : later on, up to our days, sun-dials were
constructed, and improved with the aid of* Euclidean geo-
metry. And the first step (not the last, as Dr. Campbell
thinks) was here emphatically the picking out of some
grandiose phenomenon and declaring it to go on or to evolve
" uniformly," " equably.''' Nor did these principles of chrono-
metry suffer any serious shock from the great Copernican
reform. Somehow our forefathers chose to declare the
Earth's revolution round the Sun and its spinning motion
about itsown axis as "uniform," and continued to subdivide
the associated angles. Manifestly the sun-dials, or their
prototypes, continued to show the hours in spite of the
modified standpoint. Yet these natural solar clocks had
their bad side, which perhaps is best expressed by the old
and beautiful words to be still read on some sun-dials in
Italy :
Horas non numero nisi serenas.
Other time-keepers were, therefore, invented and con-
structed in very early times, that is to say, even much before
Copernicus, whom we mentioned only incidentally â€” and in
all of them the said two features played a dominant role.
I do not propose to enumerate here all such old chrono-
metrical devices ; nor have I the required historical erudition.
But one such device attributed to Alfred the Great, who
ruled over the West Saxons (871-901), I cannot pass here
in silence, since it seems particularly characteristic in
relation to our subject. According to what my little boy
heard in his school *, Alfred the Great had good tall candles
(of what stuff I know not) made for him, and, confiding
no doubt in the uniformity of their burning down, divided
them into equal segments, and thus knew the time in day or
night. But apart from the "nisi serenas" condition, the
solar clocks had the defect of not being applicable to short
time spans (certainly not to our " seconds,"* and not even
to our "minutes"), and the other famous kind of natural
time-keepers, the human heart or "pulse/" was too often
affected by passion or disease to retain permanently the title
of uniform (here uniform succession of discrete pulse-beats).
Thus the mediaeval physicist and astronomer had recourse to
a variety of artificial chronometric devices. Even a long
time before the Renaissance complicated wheel machines
were constructed as clocks, but none of them was "well
regulated " until the times of Galileo and, more especially,
* I have no other means at the moment to verify the historical truth
of this report.
368 Dr. L, Silberstein on the
Huyghens. Properly speaking, these " chefs- d'oeuvre
d'agencement cinematique de mouvements,5' as Jules Andrade
calls them, were not '; regulated " at all? i. e. were felt not to
be worth the name of " uniformly going/' not keeping pace
with the heavenly clock. The now undisputed merit of con-
structing the first clock in the modern sense of the word is
(due to Huyghens, although it was Galileo's discovery of the
r isochronism " of small pendulum oscillations which he
utilised in such an ingenious way. Yet, before Huyghens's
invention, Galileo, who was the first to measure compara-
tively short time-intervals, constructed his own clock for the
sake of his famous investigations on falling bodies, a water-
clock that is, but more precise than the water- or sand-clocks
and the " mechanical " clocks which he inherited from his
predecessors. Galileo's own clock is, in the present con-
nexion, as instructive as the burning candle- of Alfred the
Great. It consisted of a vessel or water-basin of large
section having a very small hole in its bottom, to ensure, no
doubt, the u uniformity " of the outflow of that liquid. This
was his first care. The remainder of the procedure was
again in full harmony with our statement ; Galileo measured
the volume of the water (by weighing it, that is, but this only
to make the volume measurements more precise), and he
spoke of Â£-=1, 2, 3, etc., as proportional to the number of
equal volumes of water; this is equivalent to measuring lengths
along the axis of a well-calibrated and narrow cylindrical
vessel, if he had one. Galileo's times (the t in his great
law s-^-t2) were proportional to these volumes or ultimately
lengths, read on a metrical scale. That the same principles
can be instantly traced in all our modern clocks, watches,
and chronometers, needs scarcely to be insisted upon.
But they occur perhaps in their purest form in those
modern instruments which serve to measure very short
times, even down to one-millionth of a second, and perhaps
a little less. I have in mind Siemens's high-speed spark
chronograph. It consists, in essence, of a little revolving
drum of good steel driven by a carefully finished clock-work.
Against this drum, which we used (1897) to cover tightly
with a strip of paper blackened with a turpentine lamp soot,
is mounted an isolated platinum electrode. Sparks correlated
with the events in question pass between the platinum point
and the spinning drum and leave marks (little craters) on its
blackened surface. The clockwork is then stopped and the
drum turned round slowly by a niierometric screw, while
the marks are viewed through an appropriately placed
microscope. Their angular distance, as read on a subdivided
Measurement of Time. 369
circle of the hand-screw, gives the time-interval between
spark and spark. This is the "space-measurement"; and
the "uniformity5' was most emphatically expressed in a
letter of Siemens and Halske accompanying the apparatus,
to this effect : If you wish to obtain satisfactory results, do
not start the sparks at once but only after the drum was
already spinning for a good while, a prescription, no doubt,
based upon the makers' dynamical knowledge of the driving
machinery, but at any rate a direct appeal to what had to be
trusted to be uniform beforehand, without in this case the
least possibility of checking the uniformity by investigating,
as Dr. Campbell says in his concluding paragraph, whether
the " body covered equal distances in equal times/' the
" equality" of these minute intervals being in this case not
otherwise actually definable, unless one appealed to yet
another uniformity, viz. that of the propagation of electro-
magnetic waves along the wire-systems*. I have dwelt
upon this example, not only because it shows the two prin-
ciples in their neatest form, but also because in writing my
first paper on the time-scale, I had this spark-chronograph
incessantly in my mind. The same remarks can literally be
repeated with regard to all the familiar devices in which the
drum is replaced by a light rotating mirror used as reflector.
Very minute time-intervals are thus being measured and
give well consistent results.
Bat intervals still much shorter, the periods of light-
oscillations, are measured again on the same principles.
The propagation of light is declared to be uniform, and then
linear segments (translated more or less indirectly on a
magnified scale into an interference pattern) associated with
this propagation are measured in the Euclidean fashion.
And there is even an ever-growing a priori confidence into
the uniformity of light-propagation and a tendency to make
it the highest court of appeal for all properly mechanical
uniformities.
In short, every precise chronometry is kinematical (motion
of bodies or propagation of light), and the foremost concept
of kinematics is that of "uniform motion, " exactly so as is
that of "straight line " in geometry. Both are, theoretically,
undefined terms, and in application things are declared to
he good approximate samples of either or pointed out (with
the finger, as it were) â€” this or that is uniform, this or that
* The interval between the sparks was, in the application of the
chronograph I have in mind, due to the difference of the two corresponding
circuits, one very short and the other about 3 km. long.
Phil. Mag. S. 6. Vol. 39. No. 231. March 1920. 2 B
370 Dr. L. Silberstein on the
is straight. There is no defining of " straight " nor of
" uniform." All so-called definitions of these terms are but
apparent, each of them containing a vicious circle.
A definition of uniform motion such as Dr. Campbell
repeats after the naive little text-books [to wit : " we define
uniform motion as that of a body which covers equal distances
in equal times," p. 654] would be exactly as bad as : a straight
line is that which slopes down or up (relatively to another
straight !) by equal heights in equal horizontal distances.
It is precisely such a "definition" which prevents most
people from seeing the possibility of non-intersecting,
Lobatchevskyan straights and the hypothetical nature of
Euclid's parallel axiom. And the kinematical correlata of
these things are made manifest in my first paper, showing the
possibility of a generalized (hyperbolic) system of kinematics.
The analogy between "uniform" and "straight" becomes
still more manifest if one thinks of the modern relativist's
four-world, in which a " straight " stands for a space-straight
as well as for uniform motion or propagation. But there is
no need to appeal to that famous "union" of space and time
to show the fundamental, irreducible character of uniform
motion ; this character belongs to it historically, since times
immemorial until our days. Both the assumption of uni-
formity and the rigid subdivision of the paths or angles are
inherent in all the more precise chronometric methods ever
devised by man.
This settles the first and chief point of the present note.
It is scarcely necessary to add that in declaring such and
such a phenomenon to go on uniformly the physicist's, or
the astronomer's, choice is, among other things, based upon
reasons of convenience, aiming at a certain kind of simplicity
of laws or differential equations, such as I attempted to
explain in the introductory chapter of the " Theorj^ of
Relativity."
The second point being already settled at the very beginning,
let us pass at once to our third point, concerning that is
Dr. Campbell's own views on the question of the measure-
ment of time. Dr. Campbell is under the fatal misappre-
hension that he requires but " three definitions " in order to
set up a system of measurement of time [nay, of any other
magnitude]. These are his (1), (2), and (3), p. 653. The
second concerns only the equality of two time-intervals
whose both the beginning and the ends coincide, and the
third fixes only, in the usual way, the meaning of the sum
of two adjacent (consecutive and gapless) intervals. They
need not detain us any further. The whole burden is loaded
Measurement of Time. 371
upon (1) which reads : " The period occupied by the
happening of some definite process in a definite system is
defined to be 1" Now, if this " definite process " stood for
a single particular process covering a single interval of time,
obviously nothing could be done with (1), (2), (3). It
would amount to as much as giving on a straight a pair of
points, 0, A, calling OA the " segment 1/' and declaring
any segment LN to be the sum of the (adjacent) segments
LM and MN. This would never enable us to say what is a
segment 2, or 3, and so on.
But, if I well grasp his meaning, Dr. Campbell under-
stands by " definite process," a process such as a fall or a
complete oscillation, happening now, or five minutes hence
or to-morrow, and so on. This, however, amounts not to
solving the problem of measurement, but to massacring it
at its very root, or else it amounts to a concealed assumption
of uniformity of (in this case) the succession of a discrete
set of events. To make my meaning clear, let that standard
process be a complete oscillation of a pendulum (to-and-fro)
marked by an audible click at its beginning and at its end.
Then Dr. Campbell defines all the intervals between a click
and the next click as equal to one another, zeroth to first
= first to second = fifth to sixth, and so on. But this means
either the setting up of an indefinite series of entirely
arbitrary time-scale divisions, or else contains the tacit
belief in the uniformity of the succession of the clicks.
Such a procedure per se would not deserve the name of
chronometry ; it would be chronoscopy pure and simple.
I say, per se, i. e. without relating the pendulum-oscillations
to some fundamental kinematical and dynamical principles.
With such support the scale would ce^se to be merely
chronoscopic ; but then it would indirectly rest upon some
continuous uniform motion as the fundamental concept of
the very science (mechanics) which is its support. If so,
however, then it is preferable to utilize directly a uniform
motion (instead of a uniform succession of discrete clicks),
say, a uniform spinning â€” which brings ns back to both of
our old principles. In fact, Huyghens, who certainly pre-
ceded everybody in applying the pendulum to chronometry,
used it only as an auxiliary, intervening in his mechanism
at discrete instants, and he utilized for rigid subdivision the
continuous spinning motion of his wheels. (Such also is
the only role of the pendulum in our modern clocks.) That
our last remarks are by no means superfluous can be seen
from Dr. Campbell's embarrassment when, having dealt very
rapidly with !i integral numerals," he looks for fractional
2 B2
372 Prof. Joly and Mr. Poole : Attempt to determine
intervals. " The fractions " â€” he saysâ€”" can be obtained by
oilier pendulums," with my italics. Thus, other and other
pendulums are to be declared as fresh standards (for instance,
for t = ^ we should require a smaller pendulum, such that its
three oscillations just fill out the interval between two clicks,
for t = ^ yet another, and for what Siemens's chronograph
yields, a pendulum of ultra-molecular dimensions) ; thus the
postulate (1) would have to be extended and enriched without
any end. (Moreover, the 'fractional' pendula could only be
found by endless trials, for Dr. Campbell's set (1), (2), (3)
does not yield a method of constructing the required sub-
divisions. Nor is, of course, such a scheme adaptable to any
somewhat refined chronometry.) Is this satisfactory ? Is
such a set as Dr. Campbell's (1), (2), (3) satisfactory as the
basis, logical or physical, of a theory of time-measurement?
I think not.
Moreover, Dr. Campbell believes (1),. (2), (3) to be good
enough for a theory of the measurement, not only of time,
but also of any other " magnitude." He quotes lime only
as a little example. Now, temperature is certainly an
example of u magnitude," and better still, length or distance
is another, and it would be extremely interesting to see
Dr. Campbell setting up an intrinsic scale in both of these
cases, most especially in the latter one.
The psychological clue to all fallacies of Dr. Campbell is
contained in his concluding sentence : "Of course, this is all
as elementary as AB C." If this were so, gigantic mentalities,
such as was Cayley and many of his successors here and
abroad, would never have devoted so much time to what is
known as the Theory of Distance.
November 4, 1919.
XXXIV. An attempt to determine if Common Lead could be
separated into Isotopes by Centrifuging in the Liquid State.
By J. Joly, F.R.S., and J. H. J. Poole, B.A.I*
SINCE it has been discovered that both the Uranium and
Thorium radioactive families yield elements which
are isotopic with ordinary lead but differ from it slightly in
atomic weight and density, it has often been suggested that
common lead itself is not a homogeneous element, but consists
of a mixture of isotopic uranium and thorium lead. This view
of the constitution of common lead is based on the fact that
* Communicated by the Authors.
if Common Lead could be separated into Isotopes. ii7.9>
both its atomic weight and density are found to be inter-
mediate between those of its two isotopes, and that therefore
an appropriate mixture of the two isotopes would have the
same mean atomic weight and density as ordinary lead. If
this idea of the real nature of lead were correct, it would seem
to be possible that some separation of its two constituents,
which would differ by about 1 per cent, in density, might
be effected by centrifuging the lead while in the liquid
state. Such a separation could be most easily detected
by determinations of the density of the lead from the top
and bottom of the centrifuging tube, and this was the
method adopted in these experiments.
The centrifuge used was one constructed by Leune of
Paris, which runs at about 9000 revolutions per minute.
The lead was contained in steel tubes, which were fitted
with steel lids to a^void oxidation of the lead as far as
possible. Quartz containing tubes were first tried, but
were found too weak to stand the strain of centrifuging.
The steel tube containing the lead was heated by means of
a coil of asbestos-insulated nichrome wire wound round it.
This coil was kept in place by two collars turned on the
steel tube, one at each end, so that in effect it was really
bobbin-shaped externally. The whole tube with its heating
coil fitted into the usual outer metallic holder of the
centrifuge, which hung from trunnions in the ordinary way.
The heating current was supplied to the coil in the
following manner : â€” One end of the heating coil was con-
nected to the outer metallic holder of the centrifuge which
made contact with the main rotating spindle of the centri-
fuge through the supporting trunnions of the tube. As
it was not desirable to pass a current through the bearings
of the centrifuge, a copper gauze brush was used to make
contact with the vertical spindle of the centrifuge, and this
brush was connected to one pole of the source of current.
The other end of the heating coil was connected by a flexible
connexion to an insulated horizontal copper disk which was
fixed on the top of the vertical spindle of the centrifuge.
A vertical carbon rod was used to make contact with this
disk. This rod was insulated from the main body of the
centrifuge and held in contact with the revolving copper
disk by a simple adjustable spring device. It was connected
to the other pole of the source of current, and thus the
circuit through the heating coil was completed. As the
carbon was arranged to be concentric with the axis oi
rotation of the centrifuge, the minimum amount of power
was wasted by the brush.
374 Prof. Joly and Mr. Poole : Attempt to determine
The method of procedure in conducting an experiment
was as follows : â€” The steel tube was first carefully cleaned
and then filled to the requisite height with lead which was
melted in a small porcelain crucible. The lead used was
pure lead obtained from Johnson, Matthey & Co. The
small lid was placed on the steel tube, and the three other
carriers of the centrifuge carefully balanced against the one
con faming the steel tube and lead. The carriers were then
replaced in the centrifuge and the lead melted by turning on
the heating current for about ten minutes, after which time
the motor driving the centrifuge was started, and usually the
centrifuge was kept running for about an hour before being-
stopped. It was found impossible to run the centrifuge for
longer as the motor was inclined to overheat after this
period. When the centrifuge was stopped the lead was
removed, while still liquid, in six lots by means of glass
pipette arrangement The density of the top and bottom
portions of the lead was then determined.
The density of the lead was determined by casting small
spherical bullets from it in an iron bullet-mould. These
bullets were weighed first in air and then suspended in
methylene iodide. Methylene iodide is especially suitable
for this purpose, both on account of its high specific gravity
(about 3*3) and also its small surface-tension. Fortunately
it only attacked the lead very slightly, producing a very
slight tarnish on the surface. Some trouble was experienced
at first in obtaining sound castings from the lead, but it was
found that by allowing the mould to cool slowly from the
bottom upwards, bullets free from all cavities could be
obtained. It is essential that the conditions under which
the bullets are cast should be as nearly identical as possible,
as unfortunately the density of lead is largely affected by
the heat treatment it receives. However, the results show
that this source of error was eliminated. The balance used
was sensiiiveto yVmgrm., and the weights were standardized
against a new set by Becker & Co. which had a certificate
from the N.P.L. certifying them as correct to t$ mgrm.
Summary of Me suits.
Note : â€” W = weight of bullet in air,
B = loss of weight in methylene iodide.
w .
Hence -^ is proportional to density.
if Common Lead could he separated into Isotopes. 375
VY -:-B.
Experiment. ^ " â– â€” ^ Bemarks.
Bottom Bullet. Top Bullet.
XV 3-4015 3-4033 Top Bullet 0-05 per cent, denser.
XVI 3-4092 3-4094 Top Bullet 0-006 per cent, denser.
XVII 3-4112 3-4111 Top Bullet 0-003 per cent, lighter.
XXII 3-4090 3*4090 No difference.
XXIII 3-4085 3-4069 Top Bullet 0*01 per cent, denser.
It will be seen from these results that there is absolutely
no evidence for any separation effect. The density of the
upper and lower bullets agree as well as could be expected, as
a difference of -tV mgrm. in B which weighed about 3 grms.
would cause an error of 0*003 per cent, in the density. The
slight variation of the density from day to day is probably
due to the variation in the temperature of the methylene
iodide, and may also be caused by small variations in the
casting conditions. This point was not fully investigated as
the method is essentially a comparison one.
It is rather difficult to form an idea of the separation we
might hope to obtain on theoretical grounds, owing to our
ignorance of the equation of state for a liquid. Since these
experiments were inaugurated, howTever, Drs. Lindemann
and Aston have shown, in a paper entitled " The Possibility
of Separating Isotopes/' that if we neglect compression, and
assume equal atomic volumes for both leads, and then treat
one lead as simply dissolved in the other, we might expect
to get a concentration of thorium lead at the edge nearly
50 per cent, greater than that at the centre, if the peripheral
velocity was about 105 cm. per sec. In our case, however,
a peripheral velocity of only about 104 cm. per sec. could
be attained, which would only lead to a difference in con-
centration of about ^ per cent. This would only give a
difference of '005 per cent, in density, which is too small to
be detected by the method of determining the density used.
On these grounds, then, it is not surprising that with the
centrifuge at our disposal no positive results were obtained.
It would seem, however, certainly possible that with a
specially constructed centrifuge some definite result might
be obtained.
IveaÂ£)i Laboratory
Trinity College, Dublin.
I 376 ]
XXXV. On the Effect of Centnfuging certain Alloys while in
the Liquid State. By J. Joly,' F.R.S., and J. H. J. Poole,
B.A.I*
DURING- the course of the experiments described in
the previous paper as to the effect of centrifuging
liquid lead, certain alloys were also dealt with. The results
obtained are appended.
It will be seen that in the case of the silver-lead alloys no
definite separation could be obtained. The silver-lead alloys
were specially dealt with, as, if silver and lead could be
separated by centrifuging, the process might perhaps be
cheaper and more expeditious than the existing cupellation
method. Unfortunately the method appears to be a failure,
at least with the velocities we were able to employ.
In the case of all the other alloys an undoubted separation
was effected. In all cases, there is a considerable difference
in density between the constituents of the alloy, and no very
large amount of separation was effected. It is of interest
that a definite alloy like lead-tin alloy, which is of the
composition PbSn, should be capable apparently of being
separated by the action of the centrifuge.
Silver-Lead Alloys.
W
B*
Exp. Composition. ( *â– s Eesult.
Bottom. Top.
XIII Pb 97 per cent. 3-3761 3*3746 Positive. Top bullet about
Ag 3 â€ž â€ž 0-045 per cent, lighter.
XVIII. ... Pb 95 per cent. 3*3937 3*3947 Negative. Top bullet about
Ag 5 ,, â€ž 0'03 per cent, denser.
XXI Pb 90 per cent. 3*3892 3-3833 Positive. Top bullet about
Ag 10 ., ,, 0*17 per cent, lighter.
XXIV. ... Pb 90 per cent. 3*3922 3*3923 Negative. Top bullet about
Ag 10 â€ž ,, 0*003 per cent, denser.
Other Alloys.
XIV Pb 63*6 per cent, 2*8308 2*8129 Positive. Top bullet about
Sn 36*4 ,, â€ž 0-63 per cent, lighter.
XXVI. ... Ditto. 2-8530 2*8114 Positive. Top bullet about
1*5 per cent, lighter.
XX Pb 82 per cent. 3*0594 3-0328 Positive. Top bullet about
Sn 18 ,, â€ž 0-9 per cent, lighter.
XXV Ditto. 3-1078 3-0519 Positive. Top bullet about
1*8 per cent, lighter.
XIX Pb 32 per cent. 2*9979 2*9679 Positive. Top bullet about
Sn 16 ,, â€ž 1 per cent, lighter.
Bi 52 â€ž â€ž
Iveagh Laboratory,
Trinity College, Dublin.
* Communicated bv the Authors,
[ 377 ]
XXXVI. Notices respecting New Books.
Problems of Cosmogony and Stellar Dynamics. By J. H. Jeaxs,
M.A., F.B..S. Being an essay to which the Adams Prize of the
University of Cambridge for the year 1917 was adjudged.
Cambridge : at the University Press, 1919. 293 pp., 5 plates.
rPHE essay is a daring attempt, in continuation of previous
attacks initiated by Maclaurin, Kant and Laplace, and
followed up by Boche, Jacobi, Kelvin, Poincare, and G. 11.
Darwin, to continue the investigation of possible configurations
of a rotating gravitating fluid mass, and of its stability, and
to carry it on to a gaseous conglomeration, as of the spiral and
other nebulae.
The book falls then into two parts : in the first six chapters a
homogeneous incompressible gravitating liquid is postulated, and
the shape investigated which it can assume, starting from a
spherical form, and then endowed with rotation gradually in-
creasing which causes it to assume a variety of shape, passing-
through the Maclaurin spheroid into the Jacobian ellipsoid, and
this again into the Poincare pear-shaped figure, finally breaking
cataclysmicallv into two parts, a main body and its satellite, or the
state of a double star, the final object of Darwin's research.
In these last two investigations the mathematical difficulties
are almost insurmountable, and extraordinary approximations are
required to arrive at any definite result, and even then the
methods are not of universal acceptance, and much controversy
has been excited.
The difficulty of the existence of a free surface, and its stability,
is the chief impediment to progress : and the various stages are
very instructive in revealing the branch points where the class of
surface changes place.
Throughout the subject the angular velocity w appears involved
with the density p, in the form of the fraction 0â€” , so that o is here
the astronomical density, of the dimensions of the square of an
angular velocity, or (time)-2. Astronomical density p is converted
into C.Gr.S. density 3(g/cm3) by the factor G, the constant of gravi-
tation, G== 666 x 10-10, according to the experiments of C. V. Boys,
and then the fraction becomes ^-ftv The astronomical unit of
mass is then â€” = 107x 1-5, g, or 15 metric tons.
This fraction can be made more intelligible physically by intro-
ducing Maxwell's idea, of the grazing satellite of the' stationary
spherical globe ; then if K is the grazing velocity, and r0the radius
378 Notices respecting .Neiv Books.
K2 4
of the globe, â€” =#= -f.nQh'0 ; and if O denotes the angular
VÂ° 2 K2 4
velocity of the satellite, Â£2 = â€” T= ^7tGd, which is independent of
the radius rn : and then the fraction
b)2 li)2 _ to2 2//(jJ\2
2Â°
to which a definite physical meaning can be attached.
If the globe could retain its spherical shape when the angular
velocity was raised to Â£2, bodies at the equator would be lively on
the surface, like the mud particles on the top of the wheel of the
old hansom cab seen through the side- window, and everywhere
else the plumb-line would be parallel to the polar axis.
For the Earth this must be 170- fold, Â£2=17w.
Maxwell suggested as the universal unit of time, for the Solar
System, and ail space beyond, the period of the grazing satellite
of a sphere of water, instead of such a parochial unit as our
terrestrial mean solar day ; this new unit proves to be about
2u0 minutes.
It would be impossible to go into details here of the extra-
ordinary audacity of the mathematical attack â€¢ a mere summary
of the results must suffice, considered under the heads of
I. The Tidal Problem. II. The Eotational Problem. III. The
Double Star Problem.
Starting with the gravitating globe at rest, in the Tidal Problem
the motion is through a series of prolate spheroids : in the Eota-
tional Problem the motion is first through a series of oblate
spheroids (Maclaurin's spheroids) and then through a series of
ellipsoids (Jacobi's ellipsoids) : in the Double-Star Problem the
motion is through a series of ellipsoids.
The second half of the essay undertakes an additional difficulty
in developing a general theory of the configurations of equilibrium
of a compressible mass, in its departure from the state predicted
in an incompressible model.
Here the difficulty is great enough when rotation is absent, and
the gas is stratified spherically, and various plausible physical
assumptions must be made to allow the equations to be integrable.
Dr. Schuster's results from the limiting case of y = l'2 are of very
great importance, but a closer examination seems required to show
that the agglomeration would be unstable at the core, if a rotation
was imparted.
The object of this investigation of a compressible mass is to
frame some theory of the internal state of density in the Spiral
Nebulae visible in the telescope, conjectural Solar Systems in the
making; a feeler into Space, like Eelativity, but without abandoning
Newtonian Dynamics.
The whole essay is a direct frontal attack on impregnable pro-
blems, and will require to be reinforced by outflanking equations
of related problems that will yield to solution.
Notices respecting New Boohs. 370
Researches in Physical Optics. Part II. Resonance Radiation
and Resonance Spectra. By B. W. AVood, LLC, Professor of
Experimental Physics, Johns Hopkins University. New York :
Columbia University Press, 1919. 184 pp., 10 plates.
[Publication number eight of the Ernest Kempton Adams
fund for Physical Besearch.]
Part I of this Besearch, published 1913, was devoted to the
radiation of electrons. An adequate description in detail of
this monumental Besearch in Part II would take up more space
of the Magazine than can be spared between the multiplicity of
subjects. A mere outline of the scope must suffice for the
reader.
The author is the best known exponent of the experimental
side of the Science of Light in Physical Optics, and his contribu-
tions to the applications of Theory in recent warfare will it is
hoped be allowed to see the light now, for the general benefit of
Science.
Of the whole gamut of the light spectrum only a fraction can
be apprehended by the human eye; but the author has succeeded
in devising apparatus for picking up an impression of the part
beyond the visible rays, and utilising them in operations such as
heliograpbic work ; the signals can then be received without
attracting outside undesirable attention, as of an enemy.
We have been hearing much lately of the new Theory of
Belativity, as revealed in the Gravitation of Light, so that it is no
longer a paradox to say that Light is Heavy. In utilising the
dark rays, the author provides a discussion of the Light that is
Dark.
I he Besearch is chiefly a careful description of the delicate
apparatus required in the experimental work of the spectroscope.
Nothing is recorded that has not been observed directly, and that
is capable of being redetermined from a description of the
apparatus. No appeal is made to new theories of the aether, and
there are no elaborate mathematical developments, founded on
conjectural hypothesis ; nothing to spoil the pleasure of the
physical experimenter, and to interrupt his manipulative skill
and interest.
A Table of the Contents may be cited to show the scope of the
investigations.
â€¢ 1. Plane Grating Spectrographs of Long Focus.
2. The Besonance Spectra of Iodine.
3. Besonance Spectra of Iodine.
4. The Series of Besonance Spectra.
5. Band and Line Spectra of Iodine.
6. Zeeman-Effect for Complex Lines of Iodine.
7. A Photographic Study of the Fluorescence of Iodine
Vapour.
8. The Magneto-Optics of Iodine Vapours,
380 Notices respecting New Books.
9. The Fluorescence of Gases Excited by Ultra-Schumann
Waves.
10. A Further Study of the Fluorescence produced by Ultra-
Schumann Rays.
1 1 . Scattering and Regular Reflexion of Light by an A bsorbing
Gas.
12. Separation of Close Spectrum Lines for Monochromatic
Illumination.
Unified Mathematics. By L. C. Karpinski, Harry T. Benedict,
John W. Calhoun, Professors in the University of Michigan
and Texas. D. C. Heald & Co., 1918. 522 pp.
Perry's ' Practical Mathematics ' would be our equivalent for the
scope of this book, intended to show, here and in America, that the
old plan is obsolescent of keeping a school-boy marking time for
years over arithmetic and algebra, and then rushing him through
some Calculus and Coordinate Geometry in his last year. But
the essentials of the Cartesian geometry are inculcated here in the
use of squared paper, for drawing the simple graphs, and the
illustrations follow of the geometrical applications of the Calculus.
The logarithm is introduced at an earty stage and its use
exemplified in multiplication and division in applications of real
interest to large numbers and decimals by the aid of a compact
four-figure table. But there is no mention of the Slide Rule,
equivalent of a three-figure table, and amply accurate for ordinary
purposes. Formerly the only table to be found was a seven-figure
table, never hardly to be seen. It makes us groan when we have
occasion to turn to it, to think how late in life the use of it was
introduced to our attention.
In a first introduction to the logarithm, no base should be
mentioned except 10, and then the definition y=10*=al#,
.v=\og ?/, with the abbreviation (al) for antilogarithm, as (log) for
logarithm. A four-figure table requires the antilogarithm to make
both ends of equal accuracy, and with experience the(al) function
can be made to serve throughout in place of the (log) function.
But the heedless boy will not observe the distinction, so that many
a schoolmaster will paste down the antilogarithm table.
An antilogarithm table is soon calculated, by the ordinary rules
of arithmetic; thus y is calculated for #=0-5, 0*25, 0*75, by
ordinary square root, for #=02 by Horner's method, and then for
a?=0'l, by square root ; and no series is required.
The historical note on p. 183 seems to show that the Babylonian
clay tablet might have a different reading, as the fraction of the
illuminated part of the moon's surface is the half versed sine (hav)
of the age of the moon. We still employ the Turkish astronomy
in speaking of a new moon, and her age.
The polar coordinates, as exemplified on p. 436, should be laid
out .on the Lissajous's system, as they would be in Cartesians for
Notices respecting Nexo Books. 381
tuning-fork curves ; taking equal angular steps, and then dropping
perpendiculars on the initial line to set off: the radii of the con-
centric circles.
In a G-reekless age the Greek alphabet must not be left
undefined. And we lind no description of the Vernier or Nonius.
The whole book is very elegant and stimulating, and carries
Perry's pioneering ideas to a high stage of development and pitch
of perfection.
An Elementary Coarse of Infinitesimal Calculus. (Revised Edition.)
By Horace Lamb, Professor of Mathematics in the Victoria
University of Manchester.
Cambridge : at the University Press, 1919. 530 pp.
The large page and clear spaced print will be much appreciated
by the reader, and the diagrams are frequent enough to give
reality to the argument. The author is happily not of the school
of Lagrange, in banishing appeal to the eye in a geometrical
figure.
He does not get to work in the Differential Calculus, as under-
stood formerly, till Chapter II. A preliminary Chapter I seems
written in fear of the school of Rigour, and explains at length the
modern abstractions of continuity, sequence, convergency, dis-
continuity, and the limit, before the beginner has had occasion to
form any concept of their meaning. " Man must act first, before
proceeding to discuss the rationale of his activities."
An experienced old-fashioned instructor is likely to recommend
a skip of this chapter on to Chapter II, reserving the consi-
deration of the abstruse ideas of Chapter I till the need has
arisen in the mind of the learner. It is well not to raise a
difficulty in the mind of the beginner, until it has found a place
in his own thought.
The differential coefficient defined in Chapter II is given a
geometrical interpretation ; but the author does well to introduce
immediately another illustration, as the expression of a velocity ;
this will appeal to most minds more powerfully as a physical
realization.
The author takes a very cautious, but useful, line of treatment
in Chapter III of the Exponential Theorem and Function, and
its inverse, the logarithm : and here again he goes in fear of
attack from young Rigour, but entrenches himself very skilfully ;
making a start from the Differential Equation which defines the
function.
Applications follow in Chapter IV of the functions employed
in the course of the treatise, algebraical, circular, exponential,
logarithmic, hyperholic, direct and inverse.
Successive differentiation is treated in Chapter V, with its
geometrical application to curvature, so that Integration is not
reached till Chapter Vf.
382 Notices respecting New Books.
A daring pioneer, of the Perry type, is required to bring
Integration into its proper priority of historical order, as senior
to Differentiation by a thousand years. The idea of integration
is less abstract and easier for the mind to grasp, wheu treated
from first principles as the total growth of a quantity; instead
of its rate of growth at any instant as expressed by the differential
coefficient.
But first principles are difficult of application, and require
special treatment for each particular case, so that for rapidity of
progress in the applications, Integration is taught as a process of
Anti-differentiation, the method inculcated in Chapter VI.
The picture of the process of Integration as a Quadrature
follows in Chapter VII, explaining the operation from first
principles in its proper historical order ; and detailed geometrical
applications follow in Chapter VIII, exemplified by the calculation
of area, in Cartesian and polar coordinates, and by use of the
Planimeter, of volume and surface, of the rectification of an arc,
of mean centre, our old friend centre of gravity ; the theorems
also of Pappus, a.d. 300.
The student at this stage will begin to find himself in the
subject, after working at the carefully selected collection of
examples.
Juggling with curves (instead of letters) is the scornful name
sometimes given to the geometrical applications by the pure analyst,
anxious to get to work on his favourite detail of the Failure of
Taylor's Theorem. But where else is the student to learn
about this subject, so important for its applications in the higher
branches, except in a systematic treatment such as given here in
Chapters IX, X.
The last quarter of the volume takes up Differential Equations,
and so renders unnecessary a separate study in another volume,
of terrifying size and association, looming in store ahead.
Most of this terror could be cured, or would not arise, if the
author would introduce at the earliest stage the mere name
Differential Equation, and the associated elementary ideas. Then
the beginner could be assured that he had been working uncon-
sciously at differential equations from the start; and that the
laws of mechanical nature are revealed to us in the Differential
Equation. The separate individual problem merely assigns the
constants of the equation.
On this plan the simplest integrations would be asked for
as the solution of a differential equation ; as for instance of the
differential equations
with a drawing of the graph, and of the singular solution, as a
tac, cusp, or node locus ; and so on.
Geological Society. 383
And Differentiation would be called the formation of the
differential equation, by the elimination of the constant.
The author himself employs this idea in Chapter III, where he
proposes the differential equation -=-; =y, or Tcy,n% the definition
of the exponential function. In the associated graph the curve
has a constant subtangent.
A feature of the treatment is the banishment of Taylor's
Theorem to the end of the book. This will prevent young Rigour
from keeping his class marking time ever so long over the Failure
of Taylor's Theorem, all he seems to care about. But the beginner
is delighted with the theorem when he finds it gathers up all the
preceding isolated expansions of his functions of a real variable
in a series, and gives him a method he can employ with suitable
precautions for the numerical calculation of his function, to any
desired accuracy.
The Sublime Calculus was the former noble name of our subject,
replaced to-day by Infinitesimal. It is treated here with a view of
immediate application, as well as for the benefit of the mathe-
matician on his road to higher developments. Scores of lowly
treatises are in use, to minister to the immediate wants of the
engineer and electrician, with their presentation from the direct
commercial aspect. These all shirk such abstractions as the ideas
explained in Chapter I here.
XXXVII. Proceedings of Learned Societies.
GEOLOGICAL SOCIETY.
[Continued from vol. xxxviiL p. 748.]
November 5th, 1919.â€” Mr. Gr. W. Lamplugh, F.R.S., President,
in the Chair.
A Lecture was delivered by Hugh Hamshaw Thomas, M.A.,
F.G.S.. 'On some Features in the Topography and
Geological History of Palestine,' illustrated by aeroplane
photographs taken during the War.
The Lecturer observed that a perfectly new method of illustrating
and investigating some branches of physical geology is afforded by
Aeroplane Photography. It seems firstly to illustrate in a very
striking and. convincing form many geological phenomena, such as
the structure of a volcano or the land-forms resulting from erosion,
and may be of value in the teaching of the science. In the second
384 Geological Society.
place it ma}", in certain circumstances, become a valuable means of
research, especially in connexion with river-development or denuda-
tion in a region which is somewhat inaccessible, or where the sur-
face of the ground is very complicated and the main features are
obscured by a mass of less important detail. The lecture deals
principally with the illustration of the physical features of Palestine,
and owes its origin to the systematic photo survey made over
Central Palestine during the War. The photographs were originally
taken for the purpose of constructing detailed maps, and the
examples shown have been selected from a large mass of similar
material which still exists in the form of negatives, and these may
eventually become available in this country for further study and
research. The demarcation of the coastal plain from the foot-hills
of the upland country is often well shown by oblique air-photo-
graphs, and the weathering-out of the Hat alluvial ground by the
winter rains to give characteristic wadis is clearly seen. In the
central hill-country the terraced hills show the relation of the
scenery to the underlying rock, but their general sculpture is
regarded as belonging to a former period of great precipitation.
In arid country, where the underlying rock is laid bare, the aero-
plane camera often shows the general geological structure of the
district.
The lacustrine deposits of the Jordan Valley and their weathering
was shown, and also the form of the drainage-channels running
down into the main valley- The depression of the Dead Sea with
reference to the surrounding country has resulted in caiion forma-
tion in many places. Some evidences of faulting at different
periods can be distinguished.
The Jordan at present forms an interesting study in river-
development, and many of its main features were demonstrated.
The relation of the Jordan to the Orontes has been considered,
and an aeroplane photographic survey of the country between the
two rivers indicates that the Jordan probably originated in
Northern Syria in earlier times. The Syrian portion of the stream
has been captured by the younger Orontes, and this has had a
verv important effect on the whole topography of the Jordan
Valley.
A further study of the aeroplane photographs already taken, and
of the maps made from them, may throw much new light on the
questions of climatic changes and of topographical changes due to
faulting in Palestine.
THE
LONDON, EDINBURGH, and DUBLIN
PHILOSOPHICAL MAGAZINE
AND
JOURNAL OF SCIENCE.
[a
SIXTH SERIES.]
* April 1920.
XXXVIII. The Determination of the Rate of Solution of
Atmospheric JSitrogen and Oxygen hy Water. â€” Part II.
By W. E. Adeney, J).Sc, A.R.C.Sc.L, F.I.C., Acting
Professor of Chemistry in the Royal College of Science
for Ireland; and H. Gr. Becker, A.R.C.Sc.L, Research
'Student *.
I. Introduction.
IN the first part of this communication f a method of
studying the rate of solution of air by water was
described, and some results were given, which showed that,
when the water was kept thoroughly mixed and the water-
air surface unbroken, the phenomenon took place in accord-
ance with the general equation
dw 7
-7- = a â€” ow,
at
in which a represents the initial rate of solution, and biv the
rate of escape of the gas from the water, b being a constant
depending on the conditions of the experiment.
The method of experimenting consisted in enclosing a
laroe bubble of air, of known volume, in a narrow tube
containing de-aerated water, and allowing the bubble to pass
* Communicated by the Authors. From the Scientific Proceedings of
the Koyal Dublin Society, vol. xv. (n. s.) No. 44, Sept. 1919.
t Scientific Proa, E.D.S., vol. xv. p. 385 (1918) ; Phil. Mag. xxxviii.
p. 317 (1919).
Phil. Mag. S. 6. Vol. 39. No. 232. April 1920. 2 C
386 Prof. Adeney and Mr. Becker : Determination of Rate of
up through the water repeatedly until saturation was
reached. The pressure in the bubble was measured after
each double passage up the tube by means of a water mano-
meter, and this gave data for calculating the absorption
which took place step by step to saturation.
With the object of reducing the observations to unit area
and volume, experiments have been continued along these
lines, and the results are given in this communication. The
apparatus employed for these later experiments has been
modified in a manner which experience showed was neces-
sary ; and the determinations have been extended to include
oxygen and nitrogen as pure gases.
II. Temperature Control.
In the experiments previously recorded the temperature
of the apparatus was maintained constant by providing
a large reservoir of water, and allowing the water to run
through the water-jacket of the apparatus while the observa-
tions were being made. This method only allowed of the
maintenance of a steady temperature for a few hours, and
experiments could not be repeated at the same temperature
at will, nor could higher temperatures than that of the
room be obtained.
In order to bring the temperature under control, it was
decided to use a thermostat, and circulate the water from it
through the water-jacket of the apparatus. To provide the
circulation of water a small centrifugal pump was designed,
patterns were made and castings obtained from the Engi-
neering Department of the College, while the machining
was completed in the workshop attached to the Chemical
Department.
This pump maintained a rapid stream of water through
the water-jacket at a constant rate ; and no difficulty was
experienced in keeping the temperature constant to within
0o,l C. A farther advantage was that any desired tempera-
ture within fairly wide limits could be attained, and
experiments could be repeated as often as desired at the
same temperature on different days.
III. Experiments to determine the Effect of the Area
of the Bubble on the Rate of Solution.
(a) Measurements of the lengths of different bubbles in motion.
Bubbles of five different volumes were measured at 25Â° C.
while in motion up the tube. Arrangements were made for
Solution of Atmospheric Nitrogen and Oxygen by Water. 387
photographing each bubble and the scale in close proximity
to each other, through the water-jacket, by providing a scale
ruled on thin tracing-paper, and cementing this to the inner
tube with Canada balsam, This scale was almost trans-
parent, and the image of the bubble was orthographical^
projected on to it by means of a beam of parallel light from
an arc lamp. The camera was focussed sharply on the scale,
â€¢and the shutter was released just as the bubble passed behind
the scale, so that on the negative the scale lines were
superimposed upon the image of the bubble.
These negatives were measured by means of a travelling
microscope, and the measurements referred to the paper
scale, the errors of which were determined bv means of
a standard scale both before and after use. The length
â– of each bubble, when at rest, was measured with a mirror
scale, which was also compared with the standard scale, and
the area of the bubble was calculated from the formula
^N
v
0
^ \
V)
1
^â– ^
V,
^
^
X
v
V
v^
^
<Â£
^v
<^
â– 6 -8 1-0 12
Air-content in c.c.
Table IV.
Hesults of Experiments with Air in Bubbles of Different
Volumes at 25Â° C.
Volume.
Area.
Value of a.
Value of w0.
Value o
5 c.c.
20*52 sq. cm.
â€¢342
V59
â€¢215
75 â€ž
29-63 â€ž
â€¢450
1-64
â€¢274
10 â€ž
38-85 ,,
â€¢497
1-57
â€¢317
12-5 â€ž
47-58 â€ž
â€¢620
1-59
â€¢ -390
15-0 â€ž
56-33 â€ž
â€¢680
1-57
â€¢433
When the values of b are plotted against the area, as in
390 Prof. Adeney and Mr. Becker: Determination of Rate of
fig. 2, a straight line graph is obtained which intersects the-
axis of x in a point lying to the left of the origin. Hence
the absorption is not directly proportional to the area of the
bubble. This appears to be due to the fact that the con-
ditions which apply to the cylindrical portion of the bubble
Fig. 2.
â€¢3 *5
.c:
i-4
... .
s
^
Q/'
s
/
*
...J
50
60
10 20 30 40
Area in sq. cm.
Values of b plotted against area of bubble.
70
8&
do not hold for the hemispherical head. The rate at which
the water streams past the head of the bubble is much less
than that at which it passes down the cylindrical portion
of it ; hence the absorption due to the head of the bubble is
greater than might be expected from its area. These two
effects are differentiated by measuring the rate of absorption
for a number of different bubbles ; because, since the effect
of the head of the bubble is the same in each case, the
variation in absorption must be due to variation in the area
of the cylindrical portion. Hence the graph shows the rate
at which the value of b increases with increasing cylindrical
area. By producing the graph until it cuts the .r-axis we
obtain a constant correction for the head of the bubble,
which, when added to the calculated area of the bubbler
gives the effective area when the absorption is uniform all
over the surface.
The value of the intercept on the ^-axis is 15 sq. cm. ;
hence this amount must be added to the area of each bubble.
Solution of Atmospheric Nitrogen and Oxygen by Water.
Table V.
391
Volume.
Calculated Area.
Effective Area
5 c.c.
2052 sq. cm.
35*52 sq. cm.
7*5 ,,
29-63 â€ž
44-63 â€ž
10 â€ž
38-85 â€ž
5385 â€ž
125 â€ž
47-58 â€ž
62-58 â€ž
15 â€ž
56-33 â€ž
71-33 â€ž
IV. Experiments to determine the Effect of Tempera-
ture on the Rate of Solution of Atmospheric Air.
A series of seven duplicate experiments was made within
a temperature range of 3Â°'6 to 380,4 C. with atmospheric air
in a 15 c.c. bubble. These experiments were carried out by
the method described in the first part of this communication,
with slight modifications, and the results are given in
Table VI. and fig. 3.
Ffc. 3.
r~- 'â– â– "â– "
it
*-
5 10 15 20 25 30 35 40
Temperature in degrees centigrade.
Values of b for air plotted against temperature.
Formula :â€” 6= -0075 (T-240'l).
Examination of these results showed that, when the
temperature of the water-jacket was widely different from
that of the room, this method did not give sufficiently
accurate values for the saturation-point.
In fig. 3, the mean values of b are plotted against
392 Prof. Adeney and Mr. Becker: Determination of Rate of
temperature, giving a straight line with the formula
6 = -0075(T-240*l): This will be referred to when finally
considering the results.
Table VI.
Results of Experiments with Atmospheric Air.
Tempera-
ture,
Â°0.
Values from
log graph.
Values from
graph.
w0
Saturation Values.
Mean
Value of
b.
t.
71
b.
â€¢266
a.
â€¢750
w0.
2-720
b.
â€¢296
Dittmar.
2-534
Sum of
Readings.
2-682
3-6
â€¢281
11-4
â€”
â€”
â€¢645
2 060
â€”
â€”
2-074
â€”
11-3
54
â€¢336
â€¢675
2-115
â€¢312
2-126
2131
â€¢324
150
â€”
â€”
â€¢695
1-730
â€”
â€” .
1-806
â€”
15-0
47
â€¢307
â€¢650
1-640
â€¢338
1-988
1-614
â€¢367
200
â€”
â€”
â€¢700
1-640
â€”
â€”
1-663
â€”
20-0
44
â€¢431
â€¢640
1-500
â€¢368
1-818
1-484
â€¢399
250
â€”
â€”
â€¢720
1-570
â€”
â€”
1-573
â€”
25-0
40
â€¢459
â€¢650
1-380
â€¢411
1672
1-381
â€¢435
29-6
â€”
â€”
â€¢640
1-265
â€”
1-555
1-267
â€”
293
36
â€¢523
'660
1-320
â€¢421
1-564
1-300
â€¢472
34-2
â€”
â€”
â€¢585
1-060
â€”
1-457
1-017
â€”
344
32
â€¢580
â€¢595
1-010
â€¢406
1-453
0-934
â€¢493
38-5
â€”
â€”
â€¢630
0865
â€”
â€”
0-845
â€”
38-6
28
â€¢665
â€¢585
0-785
â€¢446
1-360
0-787
â€¢555
V. Impkovements in Methods of Experimenting.
When the form of apparatus, described in Part I. of this
communication, was employed in experiments at tempera-
tures much above or below room-temperature, a number of
possible sources of error might be expected to affect the
results, viz. : â€”
1. Difference of temperature between the water-jacket
and the air drawn in to renew bubble.
2. Absorption of air during periods of manipulation.
3. Difference in temperature between the air in the
bubble and that in the air-space of the manometer.
Solution of A tmospheric Nitrogen and Oxygen by Water. 393
Of these it seemed that No. 1 was the most important ;
No. 3 was extremely small; while subsequent experiments
have shown that No. 2 is negligible, with a narrow tube,
such as was used in these experiments.
In order to eliminate these errors it was decided to make
a new form of apparatus, suitable for use ^ith a pure gas,
such as nitrogen or oxygen; and to work with air-free
water.
(a) Preparation of Air-free Water.
In order to ensure that the water was air-free, it was
necessary to boil it in the vacuum of the mercury pump, and
then transfer it to the experimental tube without allowing it
to come in contact with the air. At first it was thought
that it would be sufficient to heat the water until its vapour-
pressure was great enough to force it over into the tube, but
it was found that this necessitated much too high a tempera-
ture. It was decided to displace the water with mercury,
but this introduced such a narrow tube between the flask
and condenser that the condensed vapour blocked it.
The difficulty was finally solved by providing a second
tube to allow the water condensed to flow back into the
flask. It was thus possible to boil the water in vacuo as long
as might be necessary, without any appreciable loss by
evaporation.
The diagram (fig. 4) shows the form of apparatus used,
and the mode of operation is as follows : â€” The water in A is
heated to a fairly high temperature by means of a water-
bath, and the mercury pump is then worked until a very low
pressure is reached. In this way most of the gas is extracted
in the first violent ebullition, and the remainder is removed
by continued boiling under the reduced pressure. During
the latter part of the operation the water bumps very
violently, with the result that some of it is thrown over into
the vessel E ; but this returns to the flask immediately by
the tube i?, as also does any water dripping from the con-
denser. When all the air is extracted, the pinch-cocks B
and C are closed, and I) is opened, when the mercury flows
in and displaces the water into the experimental tube, which
has been previously filled with mercury, and connected to
the flask A by another tube not shown in the drawing.
All the rubber stoppers used were protected from leakage by
mercury traps, and the rubber tubing was varnished with
shellac to prevent diffusion.
394 Prof . Adeney and Mr. Becker: Determination of Rate of
(b) Modified form of Experimental Tube used.
Owing to the difficulty in obtaining apparatus, it was
decided to make the required modification of the experi-
mental tube in the laboratory. It was designed and made
as shown in the diagram (fig. 5). At the upper end of the
Fio-. 4.
tube there is a hollow stopper A ground in, which controls
the connexion to the manometer D and the gas reservoir C.
This stopper also has a stopcock Ft fused on to it at the top
to permit of the tube being filled with water when used in
conjunction with another stopcock B, fused on to the lower
end oÂ£ the tube. The manometer I) and the gas reservoir C
are fused on to the tube in such positions that the openings
correspond with the holes in the stopper. The whole
apparatus is provided with a water-jacket as shown, through
the bottom cf which the ends of the gas reservoir and the
manometer project, in order to allow them to be connected
Solution of A tmospheric Nitrogen and Oxygen by Water. 395
by rubber tubing with the mercury reservoir E and the
tube G, respectively.
The manipulation involved in an experiment was as
follows : â€” After the whole apparatus had been carefully
cleaned, the manometer was filled with water and the tube
filled with mercury by connecting on a temporary reservoir
at B ; at the same time the gas reservoir C was filled with
mercury by raising E. The water-level in D was also raised
to the hollow stopper, which was then turned so as to close
all the side tubes. The tube was then connected to the
boiling apparatus just described, by a capillary tube joined
to stopcock F. The air was displaced from this connexion
by forcing a little mercury over from the temporary reser-
voir. The water was then boiled until all the air was
extracted, when the water was displaced over into the
experimental tube, by lowering the reservoir of mercury
attached to it, and raising that attached to the laboratory
flask.
When the tube was quite full it was disconnected from
the laboratory flask, and water from the thermostat was
circulated through the jacket until the required temperature
was attained. When a steady state was reached the stop-
cock F was connected to the reservoir of gas in use, and the
stopcock B to a standard burette, and the correct volume of
bubble drawn in. The gas reservoir C was then filled with
the gas by turning the stopper so as to connect it with the
bubble, and lowering the reservoir E ; the manometer space
was likewise filled by again turning the stopper and lowering
the manometer tube G. The tap at F was then disconnected
from the gas reservoir and opened and shut several times so
as to bring the manometer to zero and the pressure in the
bubble to that of the atmosphere. (The pressure in the
bubble before this operation was always slightly above
atmospheric, so that no air could enter.)
The tube was then ready for the observations which were
made as usual after each double inversion. During the
inversion the stopper was turned so as to shut all side tubes,
and it was so arranged that after inversion the bubble could
be connected to the manometer alone, to read the pressure,
and then to the manometer and gas reservoir simultaneously,
to allow of the pressure being re-adjusted to atmospheric, by
manipulating the mercury reservoir E until the manometer
went back to zero.
By means of this apparatus the difficulty about the
temperature and vapour-pressure of the replenishing gas was
overcome, because the gas was contained in the reservoir C
396 Prof. Adeney and Mr. Becker: Determination of Rate oj
at the same temperature as that of the bubble, and there was
sufficient moisture in the reservoir to keep the gas saturated
with aqueous vapour at that temperature.
When the observations were completed, the lower tap B
was connected to the gas pipette described in Part I. of this
communication, by means of a piece of rubber tubing filled
with mercury, and the water in the tube was run into the
pipette to allow of its transfer to the pump for the deter-
mination of its gas-content without exposure to the
atmosphere.
Using this apparatus, a number of experiments were made
with both nitrogen and oxygen, the results of which are
given below.
VI. Experiments with Pure Gases.
(a) Experiments ivith Nitrogen.
The nitrogen used was prepared according to a method
recommended by Knorre *, and said to give no oxides of
nitrogen.
A mixture of 30 grams sodium nitrite, 30 grams potas-
sium bichromate, and 45 grams ammonium sulphate, was
dissolved in about 500 c.c. water, and placed in a litre
retort. This was connected to three bulb tubes, the first
containing a mixture of 5 vols, of a saturated solution of:
potassium bichromate to 1 vol. strong sulphuric acid ; the
second, dilute potassium permanganate solution ; and the
third, alkaline pyrogallol.
The whole apparatus was exhausted with a water-pump,
and the liquid warmed until the pressure rose to that of the
atmosphere, when it was again exhausted, and the process
repeated. In this way the air in the apparatus was very
completely removed. The gas was collected over water
which had been boiled for some time, and allowed to cool
out of contact with air.
A series of experiments over a range of 35Â° was made
with this gas, and the results are given below.
The experimental figures were treated graphically in two
ways. In one case the rate of solution was plotted against
the mean value of the gas-content, and in the other the
logarithms of the absorptions were plotted against the time
intervals. Each set of graphs gave values for a and fr,
which are given in Table VII., and the mean of the values
of b in each case is plotted against temperature in fig. 6.
* Chem. Centr. 1903, i. p. 125.
Solution of Atmospheric Nitrogen and Oxygen by Water. 397
Table VTI.
Results oÂ£ Experiments with Nitrogen.
Tempera-
ture,
Â°C.
Values
loggi
from
aph.
Vali
les from
graph.
w0
Saturation Values.
Mean
Value of
b.
â€¢267
t.
52
b.
â€¢270
a.
â€¢644
W,
2440
b.
â€¢264
Adeuey
and
Becker.
2-230
Winkler.
2-180
Bohr.
2240
3*5Â°
111
â€”
â€”
â€¢568
1-800
â€”
1-830
1-822
1-922
â€”
11-3
44
â€¢310
â€¢558
1-830
â€¢310
1-860
1-816
1-881
â€¢310
15-0
â€”
â€”
â€¢562
1-660
â€”
1-735
1-682
' 1-785
â€”
151
40
â€¢344
â€¢533
1-640
â€¢332
1-741
1-680
1-780
â€¢338
201
35-5
â€¢403
â€¢630
1-660
â€¢380
1-556
1-530
1-625
â€¢401
24-8
â€”
â€”
â€¢646
1-480
â€”
1-476
1-416
1-490
â€”
24-8
33
â€¢435
â€¢569
1-590
â€¢394
1-488
1-416
1-490
â€¢414
30-4
32
â€¢440
â€¢600
1-360
â€¢441
1-330
1-299
1-335
â€¢440
351
29
â€¢499
â€¢608
1-260
â€¢482
1-235
1-203
1-220
â€¢490
Fier. 6.
X
5 10 15 20 25 30 35 4
Temperature in degrees centigrade.
Values of & for Nitrogen plotted against temperature.
Formula :â€” &=*00727(Tâ€” 240-8).
398 Prof. Adeney and Mr. Becker : Determination of Kate of
(b) Experiments ivith Oxygen.
The oxygen was prepared by heating potassium per-
manganate in a hard glass tube, and washing the gas with
caustic potash to remove any traces of carbon dioxide which
might be formed. The apparatus was exhausted several
times with a water-pump to wash out the last traces of air.
The gas was collected over water which had been boiled for
some time and cooled out of: contact with air.
A series of experiments was made over a range of
temperature of 35Â°, and the results treated Jjv the two
methods as mentioned in the case of nitrogen. The results
are contained in Table VIII., and the variation of h with
temperature is shown in fig. 7.
Table VIII.
Results of Experiments with Oxygen.
Tempera-
ture,
Â°C.
25
Values from ,
log graph.
Values from W0
graph.
Saturation Values.
Mean
Value of
b.
â€¢264
t. b.
77 -252
a.
1-235
w,
4-47
b.
â€¢276
A dene j
and
Becker.
4-450
Winkler.
4-600
Bohr.
4690
8-8
64 -310
1-230
3-910
â€¢315
3-814
3-970
4-080
â€¢313
15-5
59 -336
1-140
3330
â€¢342
3-254
3-372
3-450
â€¢339
202
54-5 -356
1-150
2950
â€¢390
2-970
3-040
3-110
â€¢373
25-2
51 -392
1-150
2-810
â€¢410
2-812
2-810
2-880
â€¢411
303
46 -437
1-080
2-450
â€¢432
2-485
2-510
2-575
â€¢434
351
43 -477
1-120
2-250
â€¢498
2-323
2355
2-400
â€¢487
In the above series of experiments the water in the tube
at the end of each experiment was analysed for dissolved
gases, using the extraction pump and measuring apparatus
described in Part I. of this communication. The solubilities
of oxygen and nitrogen at the given temperatures as cal-
culated from these analyses are given in Table IX., as are
also the values obtained by Bohr and Winkler by absorptio-
metry methods.
Solution of Atmospheric Nitrogen and Oxygen by Water. 399
Table IX.
Nitrogen.
Toâ„¢p* Winkler. Bohr
3-5
11-2
150
201
24-8
30-4
35*1
â€¢02139
â€¢01788
â€¢01654
â€¢01505
â€¢01392
â€¢0127(5
â€¢01183
â€¢02200
â€¢01890
â€¢01757
â€¢01598
â€¢01461
â€¢01312
â€¢01200
Adeney
and
Becker.
â€¢02203
â€¢01820
â€¢01701
â€¢01549
â€¢01456
â€¢01322
01220
Oxygen.
Temp.
Â°C.
Winkler.
A
Bohr.
I
2-5
â€¢04540
â€¢04625 â€¢(
8-8
â€¢03866
â€¢03965 â€¢(
15-5
â€¢03323
â€¢03405 â€¢(
20-2
â€¢03019
â€¢03089 â€¢(
252
â€¢02733
â€¢02805 â€¢(
30-3
â€¢02488
â€¢02552 â€¢<
35-1
â€¢02302
â€¢02347 â€¢
Adeney
and
Becker.
â€¢04390
â€¢03710
â€¢03206
â€¢02955
â€¢02732
â€¢02465
â€¢02270
Fig. 7.
! I ,Â«*-- .
1 I I
1 '
5 10 15 20 25 30 35 -40
Temperature in degrees centigrade.
Values of b for Oxygen plotted against temperature.
Formula :â€” 6 = -00672(Tâ€” 236-5).
VII. Reduction of Results to Unit Area and Volume
TO OBTAIN FUNDAMENTAL CONSTANTS.
The results have been shown to be in agreement with the
general formula
dw
^-=SAp-/
400 Prof.Adeney and Mr. Becker: Determination of Rate of
where w = total quantity of gas in solution at any moment,
S = the initial rate of solution per unit area, /= the coefficient
of escape of the gas from the liquid per unit area and
volume, A = area of surface, and p = pressure of the gas.
The values of b for different temperatures and different
gases have been found for various temperatures using a
volume of water of 101"8 c.c. and an exposed area of
71*3 sq. cm. ; hence since
we can calculate the values of /. The values are given in
the second column of Table X., and when they are plotted
against temperature in each case, three straight lines lying
very close together are obtained, as shown in fig. 8.
Fig. 8.
Â£ "4
-2
^z>
^
^
V
-P^s
?8r^
&â–
is
Â»
25
30
35
Temperature in degrees centigrade.
Values of /"plotted against temperature.
Formulae :â€” f=-0096 (T-236-5) for Oxygen.
/=-0103 (T- 240-0) for Nitrogen.
f=-0099 (T- 239-3) for Air.
Solution of Atmospheric Nitrogen and Oxygen by Water. 401
When the values oÂ£ / for any gas are multiplied by the
corresponding solubilities, the product gives the initial rate
of solution in each case, since S=/s. It will be seen by
reference to Table X. that the value of S is practically
a constant over the range of temperature given.
The value of S is approximately proportional to the
solubility, being about twice as great for oxygen as it is for
nitrogen ; and if | of the value for nitrogen be added to \ of
that for oxygen, a value for air is obtained which agrees
fairly closely with the actual figures thus : â€” J of '0083 +i of
â€¢0160 ='00665 + '00320 = -00985, while the mean experi-
mental figure is ='0100.
Temp. Â°C.
25
8-8
15*5
202
25-2
30-3
35-1
35
11-2
15-0
20-2
24 2
30-4
351
36
11-4
150
20-0
25-0
296
343
Table X.
Oxygen.
/â–
s.
s=A
â€¢373
from analysis
â€¢04390
â€¢0164
â€¢434
â€¢03710
â€¢0161
â€¢499
â€¢03206
â– 0160
â€¢545
â€¢02955
â€¢0161
â€¢591
â€¢02732
â€¢0161
â€¢641
â€¢02465
â€¢0158
â– 687
â€¢02270
Nitrogen.
â€¢0156
â€¢372
â€¢02203
â€¢0082
â€¢448
â€¢01820
â€¢0081
â€¢490
â€¢01701
â– 0083
â€¢543
â€¢01549
â€¢0084
â€¢593
â€¢01456
â€¢0086
â– 647
â€¢01322
â€¢0085
â€¢696
â€¢01220
Air.
â€¢0085
â€¢352
(Dittmar.)
â€¢02700
â€¢0095
â€¢441
â€¢02260
â€¢0099
â€¢476
â€¢02120
â€¢0100
â€¢525
â€¢01930
â€¢0101
â€¢574
â€¢01780
â€¢0102
â€¢623
â€¢01660
â€¢0103
â€¢672
â€¢01550
â€¢0104
Phil. Mag. S. 6. Vol. 39. No. 232. April 1920. 2 D
402 Prof. Adeney and Mr. Becker : Determination of Rate oj
VIII. Statement of Results.
From the figures given in the previous section it is
possible to calculate the rate of solution of the gases dealt
with, for any conditions of area exposed, depth, or degree oÂ£
saturation, provided that the water is kept uniformly mixed.
The expression can be put either in the form
dw 1
-7â€” =a â€” bu\
at
which gives the rate of solution at any instant, or in the
form iv= [w0 â€” tv{)(l â€” e~bt), which gives the amount dis-
solved at the end of any given time when w0 = saturation
value and u?i == amount of gas in solution initially. For
practical purposes it is most convenient to work in per-
centages of saturation ; hence the latter equation becomes
w= (100 â€” wx) (1 â€” e~bt), and since
>=/y
by substitution
W=(lj00--Il71)(l-
as the general equation for any given temperature, and since
/varies with temperature according to the equations
Oxygen f='0096 (T-237)
Nitrogen /=*0103 (Tâ€” 240)
Air /='0099(Tâ€” 239),
the corresponding general equation for each gas by sub-
stituting these expressions in the formulae is obtained,
thus : â€”
for Oxygen w= (100-ioJ [l-,-Â°096(T-237)^j
â€ž Nitrogen M = (100-Wl) [1 _e-â„¢W-â„¢)~tj
â€ž Air ic=(100-w1)[l-e--Â°m{T-23d)P].
As an example of the use of these formulae, consider the
question of the dissolved oxygen in 1000 c.c. water, area
exposed being 100 sq. cm., temp. 2Â°' 5 0., and initial gas-
Solution of Atmospheric Nitrogen and Oxygen by Water. 403
Percentage of Saturation.
2 D 2
w
404 Dr. A. D. Fokker on the Electric Current from
content = 40 per cent, of saturation. How much gas will be
dissolved in an hour ?
t = QQ minutes,
= 60(l-r^') = 60(l-^-222) = 60(l--8009) = 60x-1991
= 11*8 per cent, saturation*
Hence after an hour the water will have risen to 51*8 per
cent, of saturation.
These equations can also be used to calculate curves
showing the rate of solution in water of the different gases
under different conditions, and as an example the curves for
oxygen between 0Â° C. and 30Â° C. have been calculated in
percentages of saturation, and are shown in fig. 9. It is
noteworthy that when expressed in percentages of saturation,
the curves for the three gases lie very close to each other,,
those for oxygen and nitrogen being practically identical.
The authors desire to express again their indebtedness to
Dr. Hacket (Lecturer in Physics in this College) for the
interest he has taken in this investigation, and the valuable
assistance he has generously given in the mathematical
treatment of the subject.
Chemical Department,
Royal College of Science for Ireland.
XXXIX. On the Contributions to the Electric Current from
the Polarization and Magnetization Electrons. By Dr. A-
D. Fokker {Leiden)*.
AX important question in the electronic theory of matter
is the evaluation of the electric current due to the
motion of the electrons of electrically neutral atoms. To
Minkowski the idea is due to put the question as a variation
problem of a current by small virtual displacements, in a
manner to be described hereafter. Born has worked out this
idea after Minkowski's death |.
* Communicated by the Author. Abstract from a paper offered to the
Kon. Akad. v. Wetcnsch. at Amsterdam.
t Hermann Minkowski â€” Max Born, Erne Ableitung der Grundgleicli-
an yen fur die elektromagnetischen Voryiinye in beweqten Korpern, Math.
Ann. lxviii. p. 526, 1910.
Polarization and Magnetization Electrons. 405
I venture to offer a new development of the same idea,
which distinguishes itself by the extreme simplicity of the
means employed. No use is even made oÂ£ any theorem from
the theory of relativity. After completing the deduction,
nevertheless, it is easy to show the covariancy of the equa-
tions obtained in the sense of the theory of general
relativity.
In addition, one hits on a contribution of the bound elec-
trons hitherto not yet signalized, so far as I am aware (Â§7).
Â§ 1. Minkowski's Idea.
Consider a stream of neutral atoms. For simplicity's sake
we shall take them to consist of a positive nucleus and one
â– accompanying electron, both of them carrying the elementary
charge. The motions of the heavy nuclei will be identified
with the motion of matter, and we shall assume that neigh-
bouring atoms will be very nearly similar and similarly
situated, so that the functions defining the positions of the
electrons relative to their nuclei, though not strictly constant,
will vary but slowly from one atom to the next.
Of course the stream of positive nuclei will constitute an
electric current, and the stream of electrons another. As a
result of the displacements of the latter relative to the nuclei
their current will not have the same intensity as the positive
current from the nuclei. The combined effect will be
the current required in the field equations for ponderable
matter.
It will be clear that if, given the displacements, we succeed
in rinding the resulting variation of intensity of the stream,
our problem will be solved, as soon as we shall have inter-
preted the result in terms of physical quantities such as
polarization and magnetization.
The displacements can be regarded as depending on a varia-
tional parameter 6. It turns out that the terms in the result
proportional to 6 are connected to the polarization, and that
the terms proportional to 02 express the effect of magnetiza-
tion mainly.
Â§ 2. IJie displacements.
We imagine a stream of particles moving through a space
which will be described by the co-ordinates Â«r, y, z. Let
there be N of them per unit of volume, moving with velocities
clv/dt, dy/dt, dz/dt. which, after adding to them as a fourth
40G Dr. A. D. Fokker on the Electric Current from
quantity dt/dt, we shall for symmetry denote by ww, w(2\
w{3\ iv{i\ respectively. In the same way we shall often for
convenience' sake write a](1), x{2\ x^\ x^\ for a; y, z, t. The
components tv{a) are assumed to be continuous functions of
the co-ordinates and the time.
The stream-components are seen to be Nw{l\ ~Nio{2\ and
Nw(3), to which we add a fourth Nw(4) = N. They will be
altered when the particles suffer displacements as defined in
the following.
We take 6 as a variational parameter and suppose a
quaternary vector given with components r{1\ r(2\ r(3), ?*(4).
If the parameter increases by dO, the particles shall shift
from the positions (#, y, z) occupied by them at the instant
t to the positions
x + iV>d0, y + râ„¢d0, zW%)dO,
to be occupied at the instants
t + r^dd.
The components ril\ r{2\ r{'A\ r(4) are assumed to be con-
tinuous functions of the co-ordinates and time. For- each
particle the values of ra must be taken which are actually
found in the place and at the instant from where the
infinitesimal shift begins.
It will be seen that the total displacements and shift in
time of the particles from the point-instants of their un-
disturbed motions (0 = 0) will be
r(1)0, r^0, rW0, r^d,
in a first approximation, and, taking account of terms with
62 in a second approximation :
**"=Â£"(
V/A^'
rP.+ SCc)^â€” rcS
= r-e + iX(c)^d2, (a=l,2,3,4) . . (1)
where ra and ~dra/~dxÂ° have values corresponding to the
point-instants of the undisturbed motion. In this and sub-
sequent formulae the summations are to be extended over all
values from 1 to 4 of the index put in brackets.
In consequence of these displacements the stream-com-
ponents will change to
Nwa + SNioa + Â±S2Niva + . . . .
Polarization and Magnetization Electrons. 407
It will now be our business to find the first variation 8Nwa
and the second variation 52Nwa.
Â§3. The first variation of the stream.
The following conception of the stream-components will
greatly facilitate the evaluation of the variation. We keep
our eyes on the content of a space-eleme'nt dV, situated at
the point ,i,(1), x{2\ oP\ at the time <2,(4). Though physically
infinitesimal the element is supposed to contain a great many
particles so that NrfV is a great number. In an interval dt
these particles will in a four-dimensional space-time exten-
sion describe their so-called world-lines, that will fill up an
infinitesimal extension dVdt. Now sum up the components
of these lines in the direction of X", say. We find obviously
'NdV dxa. Dividing by the four-dimensional extension dV dt
we can say that the stream-component in the direction ofXais
the sum of the components described by the individual particles
-per unit of volume per unit of time :
OT da" ^ â–
-dTdT^wa'
We hardly need say that the fourth component represents
the number of particles per unit of volume. It is obvious
that these components will satisfy the condition of continuity:
2^-^r=Â° < â– W
By the displacements the component of each individual
world-line will chancre to
if we neglect 62.
The sum of the components will thus grow to
NciV f da* + 2(6)0 1~ <&Â» } .
On the other hand we must be aware that the extension
occupied by the world-lines has changed also. We can find
the increase with the aid of the functional determinant of the
408 Dr. A. D. Fokker on the Electric Current from
xa + ^xa wjth respect to xa :
B (%a +â€¢ Axa) ~d(xa + Axa)
~dxa 'dxb
d (xh + A#5) dQ6 + A^)
B#&
(dYdty
~dxa
or, consistently neglecting 02
1 + 0
d#a
d#a
1 + 0
3ra
~bxh
~drb
B^
dVtft.
-â– [i+s(^g]rfv*;
dV d*i
Thus we find after the displacement the sum of the
individual components described per unit of volume per unit
of time to be :
NÂ«.Â« + ANÂ«^= [Nw*+2(&)NÂ«m|^] . [l-S(&)0g^],
Now it must be borne in mind that this value of the new
stream-component is found in the point-instant xa-\-^xa and
not in the point-instant xa, where we wanted to know it. To
obtain the latter value we obviously ought to choose our
starting-point in the point-instant xa â€” 0ra, and take
instead of ~Niva. Thus we find, availing ourselves of the
equation of continuity (2) also :
} -)'a"NT7Â«6_r6Nwa f ,
and the first variation is
]$wÂ« + 8Nw" = ~8wÂ« + '2(b)0-^-sl r*ls
SNtoÂ« = 2(6)0^|^Nio6-rÂ»NioÂ«|. . . (3)
Polarization and Magnetization Electrons. 409
Â§ 4. The second variation.
The second variation may be found without calculation by
a formal process. Indeed, we only have to substitute 8Nwa
for Nioa in formula (3) which gives 8Nwa to get :
SB^iv^=X(b)0~1\ raS^sw^â€” r?>bl$w*~],
82mcÂ« --= X (be) 62 3t [r* ~ \ r *Nw - r^ich X
_ rbjL [ raT$wc â€” r^wa 11.. (4)
It is, however, very important to state that this formula
for the second variation implies the definition of the dis-
placements with an accuracy up to terms with 0'2 as given in
formula (1). This can be verified by a direct calculation
following throughout the same line of argument as in the
preceding section, taking account of the terms of second
order everywhere. We shall not give the calculation at full
length, we may restrict ourselves to the indication that at
the last step, viz., in choosing the right starting-point from
where the displacement will carry us to the point-instant xa
under consideration, we have to be careful to take
7 QXC
and not xaâ€”Axa, as might be thought erroneously at first
sight.
Â§ 5. The simultaneous displacements.
As yet the displacements considered have been accompanied
by a shift in time. In view of the physical interpretation
of the formulae obtained, it will however be necessary to
realise the simultaneous positions of the electrons relative to
their nuclei.
^ There is no objection to simplifying our formulae by drop-
ping 6 henceforth. Now, in a first approximation, we find
the electron belonging to the nucleus, which at the instant
x^ is in the point sP-\ x{2\ x{z\ shifted to the point
x^ + ril\ Â«Â»>+Â»#>, Â«<Â»>.+ 1*Â»;
at the instant
410 Dr. A. D. Fokker on the Electric Current from
Thus we see that its position at the time #(4) will be given by
aP + pV, ^(2) + yo('2), x^ + pW,
where
pa â€” ra â€” war(4) (5)
For an obvious reason p(4) = 0.
Next, to obtain the second approximation, consider the
nucleus at the instant
aW _ ?.(4) _ 12 / c) OL?^2OL ^Cr(4) |
when its co-ordinates are
This line implies the preceding as a special case, for a = 4.
Then the displacements oÂ£ the electron will be
f + i2(V) |^^~2(c) |^W4>,
so that its position will be given by
#a + ra_l0Â«r(4) + r(4y,4) + V((. )\Â±rc( " â€” ^a^ )
Taking a = 4 it is easily seen that this formula yields the
positions just at the instant .u(4), ?'. a},
and the simultaneous displacements are
PÂ°+i(p-V)PÂ«-ir{i){-Â£-(p-V)wÂ«} . .(a=l,2,3). (6)
Polarization and Magnetization Electrons. 411
Â§ 6. The interpretation of the first variation.
IÂ£ the negative charge of an electron, the elementary
charge, be denoted by e, the current carried by the stream of
the positive nuclei will have the components
â€” eNwa,
and the stream of accompanying electrons will carry a
current
*NieÂ« + e8Nwa -f ie82Nwa,
the resultant current from the charges bound in the neutral
atoms amounting to
eSSvP + te&Sui*'.
Now let us consider the first part, originating from the
first variation. It contains what was formerly called the
contribution from the polarization-electrons. We know by
formula (3) that
e8R wa = 2(6) -$-h -\ ei^Rufi â€” er6NwjÂ« > .
We shall consider the tensor :
pa& â€” ercq$wb _ erbT$wa.
This is the same as
= ^Sepawbâ€”^ephca,
k(7)
where pa is the principal term in the expression (6) giving
the simultaneous displacements. Introducing for the prin-
cipal part of the polarization the three-dimensional vector
we recognize in the (14)-, (24)-, (34)-components of the
tensor Yah components of polarization, and in the (12)-, (23)-,
(ol)-components the components of the well-known Rontgen-
vector which is the three-dimensional vector-product of the
material velocity w into the polarization. Collecting the
components of Tab in the scheme
[P.W]Z -[P.w]y Vx
__, - -LP-w], [p.w]x p
[p-w]y -Lp-w1x Pc'
412 Dr. A. D. Fokk'er on the Electric Current from
one may see at a glance that the first three of the com-
ponents of the variation e8Niua are the components of
rot[p.w]+p, (8 a)
and that the fourth becomes
-divp (8b)
The first variation thus furnishes a polarization-current p
and the corresponding Ront gen-current rot[p .w] ; besides it
shows a polarization-charge.
Â§ 7. The interpretation of the second variation.
The second variation too is connected by a differential
operator to a tensor, Ma& :
7\M*b
where
Mab = Â±(eraSNivb â€” erb8Nwa).
We divide this into two parts :
MÂ«6 = iÂ£?6N (r<*wb - rbwÂ«) + ^N(VÂ«Sw6 - rb8wa) ,
the first of which is nothing but a correction to the polari-
zation-tensor Vab. We can put it into the form
^SN (rawb â€” rbwa) = %e8N(pawb â€” pbu Â«),
and as in the preceding section we find contained in it a
polarization
heSNp",
and the corresponding Rontg en-vector. At a closer inspection,
taking SN from (7) we see
The latter part is in good agreement with a term of the
exact expression (6) for the displacements. The former part
implies a correction that is grasped in its meaning as follows.
Imagine that we want to know the polarization and that we
therefore choose an arbitrary closed surface, taking the sum
of the electric momenta of the atoms within and dividing by
the volume. Then our correction amounts to saying that an
atom will be reckoned to lie within the surface only when
the centre halfway between nucleus and electron lies
within.
Polarization and Magnetization Electrons. 413
Turning to the second part of M>6 we require the value of
8wa. This we get from the known values of SNw* and
8N = SNw(4> (form. (7)). It turns out to be
dt K /r B#c
Now write down
i^N (ra8ivh â€” i*8iv") = J.V)^)^-^(4)(^ - (p.V)w*)Â«* j ,
gives Â«Swi-pl8wÂ«) = ieN{pÂ»^ -p* ~\
m*(rr& -*>%),
we notice that both expressions vanish for a value 4 of one
ot: the index-numbers a or b. We recognize that
2ceV dt p dt)
nre the components of the magnetic momentum of an atom.
Thus, writing mx, my, mz for the magnetization we evidently
have
414 Dr. A. D. Fokker on the Electric Current from
In the second expression appear the quantities
\epapc,
being the quadratic electric momenta of the atoms. These
quantities figure in recent investigations of Debye and
Holtsmark on the broadening of spectral lines of luminous
gases under increased pressure "x". They seem to afford a
measure for the electrical extension of the atoms, and so it is
proposed to call their sum per unit of volume (provisionally)
the electric extension :
Kac=ieN papc.
In a form of three-dimensional vector-analysis we can con-
tract the three components under discussion into a vector
k=-[(2K.V).w],
where the number 2 added to the left of 2K has to remind us
that 2K is a symmetrical tensor and therefore (2K.V) is a
differential operator wdth vector-properties f.
This vector k is analogous to the Rontgen-vector. It
accounts in its curl for the second order influence of the
motion of polarized matter on the electric current.
Gathering the various corrections of the polarization into
a single vector n, we can collect the result of the second
variation into the scheme for our tensor Mah :
cmz+kz+ [n.w]s â€” cmyâ€” kyâ€” [n.w]y
-(3nâ€ž*k,-[n.w]2 rni + kr+[n.wl
MÂ«*(=)
-** -n> -a*
whence we get by the formula
a current :
crotm + rotk+rot[n.w] -f n, . . . (9 a)
and a charge :
â€” diVn {9 b)
We notice a polarization-current k} the Rant gen- current
* Debye, Pfcys. Zsc/w. xx. p. 160 ; Holtsmark, tf&fem, p. 162 (1919).
f Cf. the notation of Prof. J. A. Sehouten in " Die directe Analysis
zur neueren Relativitatstkeorie," Transactions Kon. Akad. v. Weten-
schappen, Amsterdam, xii. No. 6, 1919.
Polarization and Magnetization Electrons. 415
'corresponding to the complementary polarization n, and the
well-known magnetization- current crotm. As stated above
the current
rot k
originates from a second order influence of the motion of
polarized matter. It is neglected because of its smallness in
the deductions of Lorentz and of Cunningham *. In the paper
of Born cited above it is not separated from the magnetiza-
tion-current. Perhaps its action might be detected experi-
mentally if a rotating sphere of insulating material were
surrounded by a fixed circuit about its equator and placed in
a strong homogeneous electric field with the lines of force
parallel to the equator's plane. An oscillating rotation of
the sphere should induce an oscillating current in the
circuit.
Â§ 8. Remarks concerning Covariancy.
In the introduction allusion was made to the covariancy
of the result of the variation. Indeed, a reader familiar
with Einstein's theory of general relativity may easily con-
vince himself that equations (3) and (4) are invariant in the
general sense. From the definition of N?ca in Â§ 3 it is clear
that Niua is s/gx a contravariant vector, where s/ g is the
well-known factor in that theory ; for in the numerator N^V
is a definite' scalar number and dxa a contravariant vector, in
the denominator \/gdYdt would have been a scalar.
~Niva being i/g x a contravariant vector, we see that
r<*Nwbâ€”rl>l$wa'
is Vg x a contravariant anti-symmetrical tensor, and 8Nwa :
is seen to be \/gX the contravariant vector-divergency of
the tensor and therefore s/gX a contravariant vector
itself.
The same may be said, mutatis mutandis, of the second
variation 82JSiva.
Knowing the character of Fah and MÂ«6 as sj g x contra-
variant tensors, it is easy to deduce the ordinary transforma-
tion-formulae for the polarization and the magnetization.
This, however, will not be done here.
* Lorentz, Eac. d. Math. Wiss. ; Cunningham, < Principle of Rela-
tivity.'
[ 416 ]
XL. An Elementary Theory of the Scattering of Light by
Small Dielectric Spheres. By Jakob Kunz *.
THIS problem was solved a long time ago by Lord
Rayleigh f, who used spherical harmonics. In the
present analysis it will be shown that the same fundamental
result can be obtained by elementary considerations without
Fig. 1.
the use of spherical harmonics. We consider a plane polarized
beam of light, proceeding from above in a vertical direction,
Z, so that the plane of polarization is perpendicular to the
* Communicated by the Author.
i Scientific Papers of Lord Rayleigh, vol. i. p. 87 ; vol. iv. p. 397.
Scattering of Light by Small Dielectric Spheres. 417
plane of fig. 1, which contains therefore the electric force, E,
oscillating in the x direction. In the path of this beam of
light is placed a sphere of dielectric constant k and of radius R
small compared with the wave-length X. In each moment
we may consider the sphere as surrounded by a uniform
electric field, E. Now, if a dielectric sphere is placed in a
uniform field, there will be induced in the sphere a uniform
field also, and the original field outside the sphere will
be disturbed as if the sphere were replaced by a doublet of
moment
where q represents the charge in one pole and I the distance
between the two poles of the doublet. If the field is alter-
nating according to the equation e = E sin 2irnt^ the doublet
wrill oscillate according to :
/z = E^â€” ^R3sin27r^
Â£ + 2
or fju = ql sin 2 irnt
and ^ = -(27r^)2E^iR3sin27r^. . . (1)
If the doublet oscillates it will emit electric waves which
have been studied by H. Hertz. In the neighbourhood of
the doublet the oscillations are fairly complicated, but at
large distances we find simple spherical waves, in which the
electric and magnetic forces can be studied by the pulse
method as follows : â€”
The variable moment jul is either equal to
fx = ql sin 2irnt = qx
or equal to />t = gsin 2irnt .l = qv.l.
In the latter case the charge is considered variable, and the
length I constant ; in the former case the charge q is con-
sidered constant while oscillating in simple harmonic motion
through an interval 21. This corresponds to the oscillation
of a charge around a centre of attraction, and we have
da ch,! d2/jb d2x
di=qdt> d?=qd?=(i'f' â€¢ â€¢ â€¢ (2)
where /is the acceleration.
Phil Mag. S. 6. Vol. 39. No. 232. April 1920. 2 E
418 Prof. Jakob Kunz : An Elementary Theory of the
Now, if a charge having a velocity v is abruptly brought
to rest at 0, of fig. 2, a spherical pulse of strong electric and
magnetic forces will spread out with the velocity of light c.
Fig. LV
E
The ratio of the two components â€” ' of the electric force E
in the pulse will be equal to r
E,=
AB
vt sin $
qvt sin 3
ctrS
because r = ct
or
Ei!=-/ . and the magnetic rorce n. = ciht â€” â€” ^
sin 3
is vertical on the plane of the last figure in the point A. Now
let the decrease of the velocity of the charge q be Av during
At seconds, then we have 8 = cAt and
-j-, _Avq sin 3 _ Av q sinS
Scattering of Light by Small Dielectric Spheres. 419
av
In the limit ,- = â€” f, we find
at
E__/?^, (3)
t c2r . J
i /o\ t^ d2u, sin 3
or by (2) E,= - J -j- ,
or by (1) E,= E *=| W "â„¢* sin 2â„¢* ^wV
But \Â« = c, hence
jb "+ 2 r C
,-, -n^'â€” l^o47r2 sin $ . a
E, = E -^ R3 ~2- â€” sin 2â„¢*,
or, if we write es for E^ the secondary wave can be repre-
sented by
-j-,/'â€” 1 347r2sin$ . ( ,v
^ = E 7 â€” -KÂ° â€” -r - - sm 2-7ni(Â£ â€” Â« ),
where Â£'= -, ra = - ,
C A.
â„¢ /câ€” -1 -r,o 47t'2 sin $ . 2it , N , JN
2<7T /
or 0g = Es sin â€” (ct â€” r) ,
i -n T^^"~ l-no47T2 sin $
where ES = E,â€” -Pi- â€”-.s
k + J X- r
while the primary wave is represented by
Â«=Esin^(rf-Â«) (5)
For 3 = 0, the electric and magnetic forces vanish and no
light is given out in the direction x of the oscillation of the
IT
doublet. For 3 = 9 , the electric and magnetic forces be-
come a maximum on the surface of a sphere of radius r.
No light is given out in the direction x, which is perpen-
dicular to the original plane of polarization ; the light
emitted in the vertical plane yz is plane polarized ; it is
also polarized in every other direction so that the plane of
polarization in every beam is perpendicular to es (see tig. 1).
The energy per unit volume and the intensity of the light
2 E 2
420 Prof. Jakob Kunz : An Elementary Theory of the
scattered in any direction is proportional to Es2 and therefore
proportional to sin2 $.
The energy dE contained in a ring of volume
dr=2irr sin SrdS.l
due to the oscillating doublet
Es2
is equal to dE = -5â€” dr
07T
equal to dE = ^E2 (t^*gV*Â£ S-^2irr2 sin $dd,
where V= -t>-R3 is the volume of the scattering sphere.
The energy E^ in a spherical shell of unit thickness is
therefore
4
E2V2/Â£-l
or
(â€” -Y
= 3tt
but the electric energy E2 per unit volume of the primary
E2
beam is equal to Ex= â€” hence
For the light scattered by N particles per unit volume,
arranged in random order so that energies may be summed
without considering phase differences, we should have
r^Sfe1)2*' â€¢ â€¢ â€¢ â€¢ (6)
an expression first given by Lord Rayleigh by a different
method. The energy radiated from a layer of thickness dz
and of unit area is therefore
Ex -47r \*\k + 2J
or E! = E0e-
hz
VV/v-lV.
^toJ1
Scattering of Light by Small Dielectric Spheres. 421
If, however, we investigate the scattered light proceeding
in the direction z of the original beam, we have to take into
account the phases as well as the intensities of the electric
forces. Let us consider a layer dz of equal particles, of
Fig. 3.
index of refraction 7, and calculate the resultant electric
force es at any point P on the path of the original beam
where sin 3=1, due to all the doublets in the stratum. The
volume of an infinitesimal ring will be (fig. 3)
dr â€” dz 2irp dp ^=dz2irr dr
as
? + z* = r\
For sin 3=1 and E = l we get from (4)
es=
13ttV1 . 2tt. n
sm â€” {ct â€” r)*
k-r2 X
The resultant electric force due to all particles in dr is
therefore
7/ N AT 7 kâ€” 1 3ttV . 27r/ . ^2irrdr
d(es)r = N dz ] , â€” - â€” sm â€” (ct - r) â€”â€” ,
422 Scattering of Light by Small Dielectric Spheres.
and the resultant electric force clue to the infinitesimal layer
is equal to
(es)r=^dz^2 ^r I sm-~ (ct-r) dr. 2tt
or (es)r= ^
^ On the other hand, Huygens' theory gives for the path
difference
S=(7-l)Â«fc,
hence y_i = N^J-=l . . . (10)
or
v2N2(UT=^-i)2l-
If we substitute this expression in (6) and assume Ej = l
we obtain
an important relation first established by Kayleigh.
[ 423 ]
XLI. Light Scattering by Air and the Blue Colour of the Sky.
By U. W. Wood, For.Mem. M.S.. Professor of Experimental
Physics, Johns Hopkins University*.
r|^HE scattering of light by dust-free air was first observed
JL by Cabannes (Comptes Pendus, clx. p. 62), who
focussed the image of a quartz-mercury arc at the centre of
a dark box lined with black velvet, viewing the scattered
radiation through a glass window in the wall of the box on a
line formed by the prolongation of the image of the arc.
Strutt, working independently and without knowledge of the
investigation of Cabannes, observed the same phenomenon
and made a very comprehensive study of the relative scat-
tering power of different gases, its dependence upon the
density of the gas and its state of polarization (Proc. Roy.
Soc. vol. xciv. p. 453, vol. xcv. p. 155, 1918).
The results of certain investigations of atmospheric trans-
mission appear to justify Lord Rayleigh's theory that the
blue sky is completely accounted for by the scattering of the
air molecules themselves, independently of the presence of
any foreign matter.
So far as I am aware, however, no attempts have been
made, up to the present time, to compare the scattering
exhibited by dust-free air in a tube with the direct light of
the sky (as to intensity), or to compare the scattering power
of the air near the surface of the earth on a very clear day
with the average scattering power of the whole atmosphere.
Abbot's work on the absorption of light by the atmosphere
showed that the loss of intensity due to passage through the
lower mile of air above Washington was practically equal
to the total loss resulting from passage through the entire
atmosphere above the first mile (roughly four miles of homo-
geneous atmosphere). This indicates that its scattering
power is considerably greater than that of the higher atmo-
sphere, where the foreign matter is present in much smaller
quantities. There would doubtless be less foreign matter in
country locations.
The present paper deals with the scattering power of the
air close to the ground, and the photometric comparison of
the intensity of the light scattered by dust-free air, when
illuminated by a concentrated beam of sunlight, with the
intensity of the blue sky on a very clear day.
We will commence with the second subject.
The air was contained in a tube of black fibre 10 cm. in
* Communicated by the Author.
424 Prof. R. W. Wood on Light Scattering by Air
diameter and 70 cm. long, provided with two lateral tubes of
brass furnished with glass windows for the entrance and exit
of the concentrated solar beam. The further end of the tube
was lined with black velvet, which is far superior to a smoked
surface. The observation window was carried on a brass
tube 2 cm. in diameter and 15 cm. long, which was soldered
into a hole which perforated the end plate of the tube. This
was to shield the observation window as completely as possible
from light reflected from the edges of the lateral tubes. The
tube was filled with air filtered through cotton, and a beam
of sunlight reflected from a silvered glass mirror focussed at
the centre of the tube by means of a double convex reading-
glass 15 cm. in diameter. The scattered beam was easily
visible, even in a well-lighted room; it was bluish in colour,
and was practically extinguished by a nicol properly oriented.
No motes were visible. The diameter of the solar image
formed by the lens was 4 mm., which gives us an area ratio
of image to lens of 1/1400, i. e. we have a layer of air 4 mm.
in thickness, illuminated by a beam of sunlight 1400 times as
intense as normal sunlight at the earth's surface. This is
to be compared with the intensity of the blue sky near the
zenith, considered as due to the illumination of five miles of
homogeneous atmosphere by normal sunlight. It is to be
remembered that the entire atmosphere if brought to normal
pressure would form a layer five miles in depth.
As the sun was at 60Â° from the zenith at the time of the
experiment, I chose a point 30Â° .beyond the zenith for obser-
vation, in order to work with the rays scattered in a direction
perpendicular to the sunlight.
A small flake of silvered plate glass with a razor edge was
mounted in front of, and close to, the observation window.
This reflected to the eye the -light from the selected part of
the sky, which was reflected from a large silvered mirror
placed in the shade just below the mirror which reflected the
sunlight to the condensing lens which illuminated the tube.
A rotating disk of black cardboard 35 cm. in diameter,
mounted on the shaft of: a motor, and furnished with a very
narrow radial slit near its rim, was mounted close to the glass
sliver. By means of this the intensity of the sky light could
be reduced by narrowing the slit until it matched the intensity
of the scattered light in the tube. If the glass sliver is
yiewed from a suitable distance its razor edge is seen in focus
projected against the cone of scattered light in the tube,
the edge disappearing when the match is secured. This
occurred with a slit 0*2 mm. in width. The reduction in the
intensity of the sky light is given by the ratio of the slit
and the Blue Colour of the Sky. 425
width to the circumference through which it moves, in this
case 1/4867.
As we now know from Strutt's experiments, the light
scattered by dust-free air is almost completely polarized.
The light of the sky exhibits, however, not much over
â‚¬0 per cent, of polarization in a direction perpendicular to
the exciting rays. It seems reasonable to infer from this
that about 40 per cent, of its light is due to secondary
scattering (scattering of light coming from the rest of the
sky and the earth together with a certain amount scattered
by the larger particles forming the haze found at lower levels).
This means that the sky as observed in the experiment had
an intensity about 1*7 times as great as would be shown by a
column of air five miles in depth illuminated only bythe rays
of the sun, and viewed end-on, which is really what is to be
compared wTith the tube illumination. We have therefore
effected a reduction of intensity with the rotating disk 1*7
times as great as would have been required if the conditions
were as just specified.
Applying this correction alters our ratio of 1/4867 to 1/2860.
This is to be compared with the ratio calculated for the 4 mm.
of air illuminated in the tube and the five miles of air forming
the sky. Sunlight at sea-level according to Abbot's tables
has a value for the blue-green portion of the spectrum of
about 50 per cent, of its value in space, when the sun is at a
distance of 60Â° from the zenith. This has been increased
1400 times by the lens, and we can therefore represent the
scattered illumination in the tube (on an arbitrary scale),
if we call the intensity of sunlight in space unity, by
1400x4x1/2 = 2800.
We must now compute the scattered intensity which wre
should expect from the atmosphere on the same arbitrary
scale. Since we are observing a point 30Â° from the zenith,
the effective depth through which we are observing is about
1*2 times the zenith depth, or six miles of homogeneous
atmosphere. If we consider the sky as due to the illumination
of this depth of air by sunlight of its full intensity in space
(unity), the illumination will be represented by the number
of millimetres in six miles, or 9,600,000, while our tube
illumination was 2800.
The ratio of these two calculated numbers is 1/3430, while
the corrected ratio measured with the photometer was
1/2860.
The agreement is remarkably good considering the
enormous difference between the two intensities compared
experimentally, and the uncertainty about just what values
426 Prof. R. W. Wood on Light Scattering by Air
to take in the calculations. For example, the six miles of air
are considered as illuminated by sunlight of the full intensity
which it has in space. The light loses intensity as it
penetrates the air, and is reduced to about one half of its
value when it reaches sea-level. On the other hand, the
scattering power of the lower atmosphere appears to be
abnormally high, due to the presence of foreign matter, and
there is in addition secondary scattering; there is as well
probably some true absorption in the lower air. These effects
compensate to a certain extent, and on this account it seemed
best to consider the full intensity of the sunlight available
for the production of scattering, in the case of the sky.
It would be far better to make the experiment on the top
of a high mountain, or even at one of the mountain obser-
vatories, and with the data given as to the dimensions and
disposition of apparatus, the whole thing could be done in a
day or two. The easiest way to make the slit on the disk
is to paste two strips of very thin black paper on a microscope
cover-glass and then paste the whole over a larger slit cut in
the pasteboard disk. When the proper width has been found
the cover can be detached and the slit width measured with
the microscope. This was the method adopted in my summer
laboratory with very limited facilities. An adjustable slit
would be more convenient of course.
The colour match was very perfect, which alone indicates
that the light of the blue sky comes chiefly from the air
molecules; for, as will appear presently, the light scattered
by the foreign matter in the lower atmosphere is yellowish
in comparison with the colour of the clear sky.
It is perhaps open to question whether we are justified
even in considering the sky illumination as represented by
the number of molecules in the line of sight (or in other
words, the thickness of the homogeneous atmosphere) mul-
tiplied by the intensity of the illumination.
The relation holds undoubtedly for small thicknesses of
dust-free air, but Abbot's observations indicate that it does
not hold at all for the distances concerned in producing the
sky light.
For example, he finds that with the sun at an altitude of
46Â°, the sky 3Â° above the horizon is less than double the
brightness of the sky at 57Â°, though the mass of air
under observation in the line of sight for the sky near the
horizon is thirteen times greater than in the case of the sky
at an altitude of 57Â°. He shows conclusively that the intensity
of the scattered light increases rather slowly in comparison
with the increase in the number of the scattering molecules.
and the Blue Colour of the Sky. 427
In this case his value found at 57Â° was at a point of the sky
only 10Â° away from the sun, and is undoubtedly somewhat
too large, as the sky in the vicinity of the sun shows an excess
brightness due, as we shall see presently, to diffraction by
foreign matter. I have a record of one measurement made
by Mr. Nietz with an illuminometer from an air-plane at
3000 feet, which gave the zenith sky an intensity of half
that of the horizon sky.
Molecular Scattering in Directions nearly parallel to that
of the Exciting Beam.
The most elementary theory shows that the intensity of the
scattered light in directions nearly parallel to that of the
exciting beam should be only double the value observed in
perpendicular directions. This follows from polarization
considerations, for the light scattered in the parallel direction
will be unpolarized, all of the components in the incident
beam contributing to the illumination.
It is well known, however, that the light of the sky, even
on a clear day and on the top of a mountain, is enormously
brighter close to the san than in distant regions of the sky.
At sea-level on a very clear da}' the light of the sky at 45Â°
from the sun has only about 5 per cent, of the intensity shown
close to the solar disk. This estimate was made by holding
a mirror which reflected 5 per cent, of the incident light
against the blue sky, and then observing the reflected image
of the sky close to the sun, the solar disk being just hidden
by a chimney. A very good intensity match Avas secured at
about half a solar diameter from the sun, though the colour
match was imperfect, the light from the region close to the
sun appearing yellowish white in contrast to the blue sky.
The mirror was an acute prism of glass with a knife edge,
one surface being painted with black paint. The reflexion
was observed in the glass surface, and measurements with a
photometer showed that, at the angles commonly employed,
the intensity of the reflected light was roughly 5 per cent, of
the incident intensity. This device was employed subse-
quently in other experiments, and will be referred to in
future as the 5 per cent, prism (4 per cent, reflexion from
the front surface and 1 per cent, from the back).
While there appears to be no doubt but that the great
intensity of the light in the vicinity of the sun is due entirely
to diffraction by small particles, it appeared to be worth while
to examine the scattering by dust-free air in a direction
nearly coincident with that of the exciting rays. The expe-
riment was made with a tube of galvanized iron 3 metres
428 Prof. K. W. Wood on Light Scattering by Air
long and 12 cm. in diameter, painted black inside. The
ends were closed with wooden caps lined with black velvet;
they are shown in section in fig. 1, The front cap was per-
forated by a hole 3 cm. in diameter near the edge of the tube,
Fig. 1.
% VELVET
4
I LENS ".I
and a lens of 3 metres focus cemented over it. This lens
forms an image of the sun at the opposite end of the tube of
the same diameter as its clear aperture, consequently the
tube is traversed by a cylindrical beam of sunlight of constant
cross-section with sharp edges, The cap at the back of the
tube was also perforated with a hole, which was covered with
paper painted black with the exception of a narrow strip
near the edge of the hole. A hole 3 mm. in diameter was
burnt through the centre of the paper disk with a hot glass
tube. A heliostat was mounted outside of the door of the
laboratory and the sunlight reflected from a small mirror
through the long-focus lens down the tube. By observing
the unpainted portion of the paper disk the edge of the solar
image was brought up to within about 5 mm. of the smull
aperture, through which the observations were made. A
vertical diaphragm (A) concealed the lens and shut off the
glare diffracted by its edges and small scratches on its
surface. It was thus possible to bring the pupil of the
eye up to within 5 mm. of the cylindrical beam of sunlight,
and look diagonally across it down the tube at the black
velvet background, the direction of vision making an angle
of less than one degree with the light-rays. Before the
introduction of filtered air the amount of light scattered
along the path of the beam was considerable, each mote
shining with dazzling brilliancy. With filtered air the
velvet background appeared quite black, but the residual
luminosity of the intervening air was at once apparent when
examined with a simple piece of apparatus, which, for
want of a better name, we may call a nigrometer. It is
merely a tube of pasteboard 3 cm. in diameter and 40 cm.
and the Blue Colour of the Sky. 429
long, painted black inside, with one end covered with a cap
of thin black paper perforated with a hole 3 mm. in diameter.
If the end of the tube is pressed into the socket of the eye
and the tube directed towards some object, the blackness of
which we wish to test, the hole is visible as a more or less
feebly illuminated disk, unless the object is absolutely black,
in which case the hole is invisible. It is advisable to have
an extension tube projecting beyond the cap, otherwise the
edge of the hole may be seen illuminated by diffracted light.
This device is so sensitive that it will show the scattering of
light by one foot of air on a clear day, illuminated by normal
sunlight : in fact, I have detected the scattering of 1 foot of
air illuminated by the light of the blue sky ten minutes
before sunset. As a background I employed a black cave, a
wooden box 1*5 X 1 x 1 metres, with a hole 40 cm. square at
the end, painted black inside and furnished with a curtain
of black velvet at the back. This box is placed out of doors
in the sunshine, the sun's rays being parallel to the ends. If
the nigrometer is brought close up to the aperture, the hole
at the end of the tube disappears, coming into view as a very
pale bluish disk when we move back about 30 cm. If we
look at the open window of a house 1000 feet away on a clear
day it appears extremely black, and if we imagine this to be
reduced to 1/1000 of its intensity we should expect that the
residual light would be unappreciable by the eye, yet this is
what we actually see with our black cave at a distance of
one foot.
Coming back now to our experiment with the long tube, we
find that if we observe the interior with the nigrometer the
scattered light is very conspicuous. If, now, the scattered
intensity in directions nearly parallel to the exciting beam
and in directions at right angles to it are as 2 to 1, the long
tube should show an intensity twice as great as a stratum of
air 3 metres in depth, traversed by sunlight in a direction
perpendicular to the line of vision. This comparison was
made in the following way on a very clear day. The dark
cave previously referred to, was placed at a distance of
3 metres from the open door a little to one side of the
heliostat. By means of a mirror and a silvered sliver of glass
placed in front of the observation hole of the long tube, a
photometric comparison was made without difficulty. All
windows in the laboratory were darkened, and screens of
black cardboard were mounted as required to exclude un-
necessary light ; the aperture of the door was also contracted
with curtains and screens. It was found that the luminosity
in front of the dark cave was at least twice as bright as that
430 Prof. R. W. Wood on Light Scattering by Air
shown by the dust-free air in the tube. This is about; what
we should expect, as the outside air was not free from
foreign matter, and as will appear from experiments to be
described presently, the scattering power of the air close to
the earth's surface is from two to three times the average
scattering power of the atmosphere taken as a whole. The
colour of the luminosity in front of the dark cave was
decidedly whiter than that seen in the tube, which also is to
be expected. This experiment makes it appear probable
that if the atmosphere were absolutely free from foreign
matter, the sky would be no brighter at the sun's edge than at
remote distances*. The performance of the long tube was
very satisfactory : it was made from oddments from the junk
heap, and the lens ground and polished from a piece of thin
plate glass. The silvered glass slivers referred to are made
by silvering a piece of plate glass, polishing the surface, and
then breaking off thin scales by striking the edge with a
hammer in a direction parallel to the silvered surface. The
razor edge oE one of these scales disappears when the photo-
metric balance is secured quite as completely as the dividing
line of a Lummer-Brodhun prism. The observation hole was
left clear, as it was found that a glass cover was apt to send
some light to the eye.
Scattering of Air close to the Earth's Surface*
This question was investigated by determining the depth
of the stratum of air in full sunlight (observed against a
black cave), necessary to give a luminosity equal to 5 per cent,
of the luminosity of the blue sky 60Â° from the sun on the
clearest days.
A 4'5-inch astronomical telescope was directed towards
the black cave and the eyepiece removed. The real image
of the aperture of the cave was then examined with the
nigrometer, the 5-per cent, reflecting prism being held in
front of the small circular hole in such a position that its
thin edge bisected the aperture and reflected the light of the
blue sky to the eye. On slightly hazy days an intensity
match was secured with the cave at a distance of 400 feet,
while on the clearest day of all the distance increased to a
trifle over 1000 feet. Similar results were secured by observing
the open doors and windows of distant buildings with the
nigrometer without employing the telescope.
In discussing these results the question again comes up as
* Dr. Luckiesli and Mr. Nietz "both report having frequently observed
that at great altitudes (15,000 to 20,000 feet) the sky appears dark blue
right up to the edge of the solar disk.
and the Blue Colour of the Sky. 431
to the value which we are to assign to the intensity of the
sunlight effective in illuminating the five miles of homo-
geneous atmosphere, if we consider the intensity at the
earth's surface as 50 per cent, of its value in space. If we
make the same assumption as before, we should expect for a
very clear day 1000x1/2 = 5 per cent, of 26100x1 if the
atmosphere were uniform in composition. To have this
equation hold we must multiply the left-hand term by 2*64,
which means that the scattering power of the air close to
the earth's surface is about 2 6 times the average scattering-
power of the entire atmosphere.
These results appear to be in good agreement with Abbot's
observations on atmospheric absorption (absorption in this
case to be understood as the removal of energy from the
primary beam of light by scattering). Abbot found that the
loss of intensity due to passage through the lower mile of the
atmosphere was equal to the loss suffered by passage through
the entire atmosphere above the first mile. He also found
that the intensity of the solar radiation for the middle of the
visible spectrum at the earth's surface was about 50 percent,
of its value iu space. If_, now, we consider the atmosphere as
tin ocean of air at standard pressure five miles in depth (the
O-mile homogeneous atmosphere), and if we consider the
percentage of foreign matter as constant throughout its mass,
each mile will remove 12 per cent, of the energy from the
light, and passage through the entire five miles will give a
residual intensity of 53 per cent, of its original value. This
12 per cent, we may call the average absorbing power of
one mile of the atmosphere, the quantity which we are to
compare with the scattering power of 1000 feet of surface air
observed with the dark cave. The facts of the matter are
that most of the foreign matter is in the lower mile. Con-
sidering the atmospheric ocean as divided into two layers, a
lower layer one mile in thickness and dust-laden and an
upper layer of four miles free from dust, an absorption of
-30 per cent, by each layer will be in close agreement with
Abbot's results. This absorption of 30 per cent, by the
lower mile is 2*5 times the average absorbing power of one
mile of the homogeneous atmosphere, and is in good agreement
with the observation that the scattering power of the air near
the ground was 2'G times the average scattering power of
the entire atmosphere.
Demonstration Apparatus for the Laboratory.
The phenomenon of light scattering by pure gases is of
such fundamental importance from a theoretical standpoint,
432 Light Scattering by Air and Blue Colour of the Sky.
that apparatus for its exhibition to students should be in every
physical laboratory. For demonstration purposes and for
the investigation of rare gases, which it is difficult to keep
pure in metal tubes with cemented windows, the following
form of apparatus will be found superior in many respects
to the branched metal tubes employed by Strutt in his inves-
tigationsi The tubes are made wholly of glass, and can be
hermetically sealed, insuring the continued purity of the
gas, and they can be set up and exhibited at a moment's
notice. They are prepared in the following way : â€”
A glass tube about 25 mm. in diameter and 25 or 30 cm.
in length, is drawn down in an oblique direction at each end,
as shown at A, fig. 2.
Fiir. 2.
The flame of the blast-lamp is then directed against one
end as shown in the figure, and the end blown out into a
bulb, as shown at B.
This method of preparation obviates the necessity for
sealing in lateral tubes for the admission of the gas, and will
be found a great saver of time in the construction of all tubes
for the study of the optical properties of gases and vapours.
The other end of the tube is left as shown at A in the figure,
as the sloping wall obviates the back reflexion of light from
the bottom of the dark cave which this end of the tube is to
form. Two lateral bulbs are next blown out from the wall
of the tube : these should be as large as possible consistent
with strength. Two tubes should be prepared, one filled with
air filtered through densely packed cotton and the other
with ether vapour, which has over twenty times the scattering-
power of air. The air can be forced into the tube through
the filter with a tire-pump, and the small lateral tubes
sealed-off with a flame. A small quantity of liquid ether is
introduced into the other tube, and the entire tube warmed
by sweeping it with a bunsen flame, or better by immersion
in water heated to a temperature of about 40Â° C. The jet
of vapour which escapes from the small tube can be ignited
after waiting a few moments for the expulsion of the air.
As the tube cools the flame dies down, and just as it is on the
point of going out the small tube is sealed. The tubes are
An Alternative View of Relativity. 433
now painted with thick black paint, with the exception of
the dotted portion oÂ£ the three bulbs. They are mounted in
a horizontal position, and sunlight, reflected from a mirror,
is focussed at the centre, with a 6-inch reading-glass, the
light entering through one oÂ£ the lateral bulbs and passing
out through the other. Observations are made through the
clear portion of the bulb B. A nigrometerwith a hole about
8 mm. in diameter will be found of assistance in viewing the
luminosity of the gas at the focus, though in the case of ether
it can be seen at once even in a well-lighted room.
Summary.
1. The intensity of the light scattered by a given thickness of
dust-free air in a tube illuminated by concentrated sunlight,
has been compared photometrically with the light of the sky,
by reducing the intensity of the latter until a match was
secured. The ratio of the two intensities was compared with
the calculated ratio, making certain assumptions in the case
of the light of the sky, and a fair agreement found.
2. The intensity of the light scattered by dust-free air
nearly in the direction of the incident light has been examined
and found to be not very different from the intensity scattered
in a perpendicular direction. It is theoretically twice as
bright, but the conditions of the experiment did not permit
of the determination of a difference of this amount. This
indicates that the enormous increase in the intensity of the
sky close to the sun's limb (over 20-fold) results from
diffraction by motes in the air, and would be wholly absent if
the atmosphere were perfectly clean.
3. The scattering power of the air near the ground on the
clearest days in the country has been found to be about
2' 6 times the average scattering power of the atmosphere.
XLIL An Alternative View of Relativity.
By Prof. Fkederick Slate, University of California *.
MANY expositions of relativity are marred at two points
in the cogency of their aggressive argument for
rejecting Newtonian dynamics. Where their reasoning is
based explicitly or by implication on a statement of Newton's
second law for constant inertia, supported by a " Principle
of vis viva " requiring all work to be recorded as change of
kinetic energy, these two restrictions foredoom to failure the
* Communicated by the Author.
Phil May. Ser. 6. Vol. 39. No. 232. April 1920. 2 F
434 Prof. F. Slate on an
attempt to use that narrowed Newtonian scheme in electronic
dynamics. Is it not one typical or even unique function of
the electron to effect reversible transformations between
mechanical energy and field-structure ? Restoring duly
general form to Newton's relations for energy, momentum
and force, leads immediately to an adjustment of relativity
and Newtonian dynamics, eliminating alleged contradictions
and re-establishing them as two properly equivalent pro-
cedures. Let us proceed to exhibit and confirm this, limiting
the discussion, however, to the simplest case as a conclusive
sample : An electronic particle in progressive rectilinear
motion under electromagnetic field-forces.
Assuming a reference-frame (F), the tangential force (T)
when inertia (???) is variable becomes
T_ d . v do dm /1X
1= -r [mv) = m -=- + v ~j~ (I)
dt v ' dt dt w
Forces of this type are not invariant for frames (U) having
unaccelerated translations (u) colinear with (v) ; the so-called
"Newtonian transformation " loses validity. For any such
frame an apparent force (rJY) will be determined in relation
to the observed force (T0) by
m f dv , N dm m . dm m ,_N
!-,â€ž_+(â€ž_â€ž)_. T.'+.^-T.. . . (2)
The second equation here is significantly parallel with one
connecting gravitation and weight. At the equator these
are colinear, and
^-m^G)2^7,; W1 + m1rofi=Gtl. . . (3)
Continuing, write for activity (A) and work (W)
a_ t dv â€ždm d' . 2 dm , ..
A = t?T = m^+tr-i-=^(imWÂ»)+^w; . ... (4)
W = CvTdt= f ^Umv2)dt+ fw~dt
Jo J odt Jo at
= [E]+i|\.25
V2 V~~~c*
Consequently if, without departure from c.G.S. units, we
define a new variable speed (vc) for (m) and recalculate :
, v â€” u v' Vc +n
Vc =
uv 1 uv .. uvc
1? i~'7 + c2
i-5 (i-?)(1+'f)
it follows that
Alternative View of Relativity. 437
In this aspect of "Einstein's theorem," which equations
(13) in effect reproduce with altered meaning, it furnishes
a rule for making compensations in activity, for disturbed
value caused by passing over to a new frame (U). It is
self-evident how the procedure can be reversed, correcting
thus a distorted estimate in (F) of observations (or measure-
ments) made in (U). Reading then the established rule for
superposition of colinear " Lorentz transformations" in this
novel sense, any linked succession of repeated distortions
from original data can be traced through our frames (U),
and the net compensation at any stopping-point determined.
The proof is simple that the net effect (disturbance) is nil
whenever the chain is a closed one.
Passing from activity to a similar analysis for tangential
force, differentiate the first of equations (13) as a beginning.
This gives
S=k)(i-3]1(V).. . . d5)
Hence, quoting the easily proved connexion
M^h^> â€¢ â€¢ â€¢ ^
= 7F)7(k0^" ' (17)
We are brought thus to what is formally identical with
the " transformation of Minkowski-f orce " (K) :
Y(Â»)(i-"-)t, = (7(â€ž)T:);
[T.=K(F); 7(Â«)T;.3K'(U)}, . (18)
not overlooking that Minkowski's "proper time" (and not
" local time") replaces fluxion-time (t). All of this illuminates
vividly the corresponding statements according to the method
of relativity, and is readily seen to put in hand a complete
control for a new aim of the whole system of calculative
detail that flies the flag of "non-Newtonian mechanics."
The particulars of that restoration to the older allegiance
need not concern us here, beyond showing how the work-
equation is equally tractable.
Let the interval begin at (v=zu). Then for the apparent
438 An Alternative View of Relativity.
manifestation in (U) of work associated with a process in
(F), we have after a short reduction,
jV-^fcA.^fl-g-^]
= <^(yK)-1), â€¢ (19>
' (v,)
and if v is taken along O.r this gives
(tEJ)*^1-1'"*) (vL)
This is the well-known and important transformation of
(l-u/2).
From (iii.) by making use of (vi.)
! Ux â€” V
1 â€” VUX
442 Mr. H. T. Flint on the Applications of
Similarly, we may derive
11 V 1
and uz
y 0(1 -vux) "* 0(1 -vux)
by taking v along Oy and Oz respectively.
3. Let m0 denote the " rest mass " of a particle, and then
Jsl = m0io
is also a physical quaternion â€” the momentum quaternion.
We obtain from Â§ 2
m0 m0
0(1- vux), . . . (vii.)
(l-u'*)i (1-U2)i
or writing this
m' = m0(l â€” vux)
we obtain the usual transformation for mass.
4. By a second differentiation we pass to acceleration and
write :
- dw
Transformation of acceleration is completely expressed by
/'=Q/Q.
5. We write P = w0/
and call P the physical force quaternion.
We then have
(cW dH2\ +>, ~ (dH d2l\â€ž
Thus
d2v' dH d2v d2v
mÂ°dr2- =imol3v7 ^ +â„¢Qâ€”2 + v(l-Â£)ni0S^-j . v, (viii.)
and
d2l' Q> +niQipvb -T-f. v (ix.)
If v is along Ox we find from (ix.)
dm' dm mv dux
dt' dt (1 â€” vi(x) ' dt
(X.)
Quaternions to the Theory of Relativity. 443
Writing Fx= ~r(m0ux) etc. we find from (viii.)
at
-^ dm 1
hxâ€”v â€”
F,'Â« ^ â€” =Far- 1-^- Fy- ^-^^F,
1â€” tnix 1 â€” vux y 1â€” vmx )>.(xi.)
and similarly,
F '= -^ F ' =
y /3(l-iO' * ${l-mix) j
These are Planck's equations for transformation of force.
6. If mass be regarded as a manifestation of contained
energy we may, on this view, regard m0 as a measure of
the energy of a body at rest.
The expression for the energy is â€” â€” ^\i â€¢ Thus the
dl . * [Â± â€” u)z
scalar term m0 â€” of M is equal to i (energy) .
From the definition of dr we have
(dT)2=(dvy2-(di)2 (xii.)
Hence /dv\2 (dl
an
or
d
dr\dr) ~~ dr\dr) '
0 dx d2v dl dH , ... ,
Multiply throughout by m0 and the term on the left becomes
1 d_ f dr)
(i-^2) dt\mdtj'
On the right we have
1 d f m0 )
(1-m2) ^ ( (l-tiÂ»)*J '
Thus
*dt 'dt'\m7t)~ dt' X (l-u2)i J
This equation represents the principle of conservation of
energy, for on the left we have the activity of the force and
on the right the rate of change of energy.
444 Mr. H. T. Flint on the Applications of
If Kitf denotes the quaternion conjugate to iv and P is the
force quaternion, the equivalent of (xiii.) is
SP.Kto = 0 (xiv.)
This is the same as the condition for constancy of internal
energy given in the ' Theory of Relativity ' *.
The quaternion Kw or any physical quaternion of the form
is transformed to S/ by the operation Qc^Qc, where Qc is
derived from Q by writing â€” v instead of v f.
Transformed to S', SP . Kw becomes
SP'Kâ„¢' = SQPQ . QcKioQc=SQPKk;Qc.
Thus PKio is an " P " quaternion J whence its scalar is
invariant. Thus SP'Kw'=:0, or the principle of energy is
invariant.
7. From (xiii.) we obtain a more general result by
regarding the term on the right, viz. -=- -t~(~t ) prefixed
with the negative sign and multiplied by m0 as the rate of
change of energy, i. e.
~r (ener
oil d [ dl\
This leads to the expression ^m0k2 for the energy, omitting
an arbitrary constant.
We may denote the kinetic energy by the expression
It has been pointed out by Jeffreys || that while there is a
certain arbitrariness in the choice of the exact form for the
kinetic energy there is convenience in the adoption of this
form.
This expression, like m0(kâ€” 1), reduces io the ordinary
value \mv2 for velocities very much smaller than that of
light.
* Cunningham, Theory of Eel. p. 167.
t SilbersteiD, Phil. Mag. May 1912.
% Silberstein, ibid.
Â§ Cf. W. Wilson, Proc. Phys. Soc. xxxi. pi. ii. p. 74.
r| H. Jeffreys, Phil. Mag. July 1919.
Quaternions to the Theory of Belativity. 44 5
8. The equation of motion is to be written
where P = F + A and the relation between the scalar and
vector parts oÂ£ P is
Â£KA=S~F,
dr
and this is the same as
skâ„¢p=o.
9. An examination of P shows that it is constructed
so that
F=Â£p and A = ik-y-,
where p is the force as it enters into ordinary mechanics,
and -=- is the rate of change of energy.
We may easily derive the force in S' in terms of the
S measure. We have merely to transform P',
P'=F'+A' = Q(F + A)Q.
Equations (i.) and (ii.) give immediately
F' = A(l-Â£2)u + F + v(l-/8)SFv . . (xiv.)
and A' = Â£A + (l-Â£2)*SFv .... (xv.)
at
Using the ratio p given by (v.)
p-rv(l-/5)Spv-^v^
i at / â€¢ \
P = 571 o a > â€¢ â€¢ (XV1-)
and in the same way
dio ~
dt' " 1 + vSuv *
(xvii.)
These two equations represent the general transformation,
and there is no particular direction for the vectors occurring
in them. Equations (xi.) are particular cases.
As an example, we may apply the transformation to the
mechanical force on a moving charge.
446 Mr. H. T. Flint on the Applications of
Thus if _ ,T _ dio p.â€”
p=E + VuH, ^-=â€” SEu,
and making use of the Principle of Relativity, the physical
laws being unaltered by transferring to S'.
We have
p' = E' + V(yiT), ~ = -SE'u'.
On making the appropriate substitutions in (xvi.) and
(xvii.) and remembering that
, u + v(l-/3)Suv-/3w , ... N
U = -o/i o N â€¢ â€¢ â€¢ â€¢ (xvin.)
We find
E'+v(I-Â£)SE'v = /9(E-tEJ, H/=Â£(H, + t?E,),
or H' + v(l-/3)SHV=/3(H + ^VEv). . . (xx.)
10. The Field due to a uniformly moving electron.
The case of the uniformly moving electric charge can be
easily dealt with by means of equations (xvi.) and (xvii.).
The problem is to determine the field at a point in
system S due to a charge moving with velocity vv. If the
system S' moves with this velocity the charge is at rest in
that system, and from the point of view of 8' observers the
case is electrostatic.
Consider a charge, e, at rest in S' and suppose there is a
unit charge at a point P' moving with velocity u'. We shall
ultimately write u'=â€” vv, so that P' is at rest in S.
The force on P' is
. e i â‚¬ .
p' = 7j - . ri = -,3 . r',
where r/ is the unit vector in the direction from e to P'.
Let e be situated at the origin for convenience. Also
dw' e ,
where r'% means the cube of the tensor of x' .
Quaternions to the Theory of Relativity. 447
Thus from (xvi.) by applying the transformation from
S' to S, i. e., writing â€” v instead of v in the formula
^^r=W)^-{r'+v(1-/3)SrV-^vS,lV}'
or writing u/= â€” vv and after a simple modification
P=S{*'+*Sr'v(l-J)j. â€¢ â€¢ . (xxi.)
It is, ol course, natural to measure from the instantaneous
position of the moving charge e, as it is viewed by observers
at rest in S. It is easy to take this new point of reference.
For let the instant in S' be zero, i. e., t' = 0. From (i.)
r ' = - v0Yt + r + v(l â€” /3) Srv,
and from (ii.)
These are merely the Lorentz-Einstein formulae.
If t' = 0,
r^r-f vSrv
I1- 1)
If 0 is the initial position of the electron, i. ) combinations. If,
however, the weak line rt=(l)49069-64Â±2-4 be taken as
F1 (*2j with P2 (2) not observed, the resulting formula gives
the Spectrum of Copper. 461
lines for m = 3....6 whose wave numbers are 55031-92Â±1*30,
57710*49Â±*75, 59141*05Â±*5, 59994-15 in which the am-
biguities are due to possible errors in Px (2). These all lie
in the ultra-violet region of Handke. In comparing observed
with calculated it should be remembered that Handke gives
his measures to *1 A., whilst dn varies from 28 d\ to 40 dX.
Consequently any value of dX between +'05 is equally
probable or of dn between limits varying from 1*4 to 2. In
addition are observation and standard errors of which no
estimates are given. The formula for P: is
| m + -891742- * \ .
m = 2. 49069. This gives Pl (1) -Pl (2) == 18286*87 Â±2*4. This
is not observed, but an observed line n= (2 n) 18475*93
treated as p2 (1) â€” p2 (2) gives the P (2) separation as
59*38 zb, the expected amount.
m = 3. [55031-92 + 1-30]. There is an observed line at
(lj 55026-69-30 dX. Also (2 n) 24245*24 (*6) as
^i(l)-jÂ»i(3) gives P1(3) = 55028*01Â±'6.
m = 4. [57710*87 + *75]. There is an observed line at
(4) 57723*4- 33 dX.
Also (2n) 26931-74+-3 asp1(l)-p1(4) gives
P1(4) = 57714-51Â±*3.
(2 n) 27188*69 as p2{l) -p2 (4) gives
P2(4) = 57723-23Â±-4.
The former supports the calculated P1 (4). The second
is the observed, but it is curious that P2 should be seen
and Px absent.
3n = 5. [59141-05zf5J. This is not observed, but it is
connected by it-links to lines on both sides. Thus
u. (8)59822-9 = 59142*24-36^,
(6) 58459-0. u = 59139-70-34cZX,
an exact series inequality.
Also (2n)28366-38 as j>i(l)-jpi(5) gives I\(5) =
591 49*15 db'8, probably P2 with no observed Px as in
the previous order.
m = 6. [59994*15]. This is not observed, but there are
r-linked lines for Pls P2, viz.
(6) 59297-9 . v = 59989-9-35 dX
(8)59290-9 .v = 59982-9-35 rf\.
These would give theP(6) separation as 7 â€” 35 (d\iâ€”d\2)
= 7+3*5 as against a calculated value of 4. It thus
aorees within equally probable errors.
Also [2n) 29212-61 as Pl(l)-i?1(6) gives Px(6)
= 59995*38-h*8 â€” also to the violet of the linked lines.
462 Prof. W. M. Hicks on
The next two calculated lines are 60543, 60917. There
are strong lines at 60536-36*6dX and 609087-37d\,.
but they are too strong and moreover show evidence of being
summation lines (see later, p. 473). The formulae reproduc-
tions combined with the observed combinations would then
seem to support the allocations given as at least one of the
frayed out fragments of the system. The whole region round
the observed combinations is full of lines of the same nature
(2n) and may possibly be combinations for some of the
other P fragments. The combinations considered above are
collected in Table III. at the end.
The establishment of the hypothesis of the break-up of a
normal series into a large number of displaced, and linked or
otherwise shifted lines is of fundamental importance. The'
laws regulating this break-up can only be discussed when a
large mass of material for comparison has been collected.
As a contribution to this some instances are considered in an
appendix.
Spectral constants. â€” With the establishment of the S limit
as 31523*48 + fit becomes possible to apply the same methods-
as were employed in the discussion of the spectra of Ag and
Au to determine the value of v more accurately and to
deduce therefrom the value of the onn and of the various
links. There are no interferential measures giving v directly
but, as in Ag, Au, Fabry and Perot give such for Dn(2) and
D22(2). K. R.'s values are very accurate and give definitely
that A = 50 8 and that the satellite separation for D(2) is
due to 236V The separation of Dn, 1)22 given by F. P/s
measures, \\ 5153*251 4- -001.^, 5218*202 + *001^ in LA.,,
is 241-4632- -00376 #2 + "00367 a?!. The calculation carried
out on the same lines as in the case of Ag and Au gives
in R.A.
v = 248-44402 - -0038 fa - a*),
A = 7307-087--3310f--113ff2 + \L10a?1,
8 = 146-1419 -â€¢00662f-'0022(3) = 6876*08 ; ihe lattev
/a = Â£885-8 -31 d\.
The separation of fi(3), /2(3) appears as 3*54, 3*15, but
the measures are not very exact. A displacement of 6^8
gives a separation 3" 14. wi="4." electric field strengthened it, type bination. stark's measures zero used. i(4)="4402-00," ,2(4)="4399'56." (4)="4402-00," 2(4)="4399-68." difference sum- mation thus agree great exactness. they mean ^i(l), 31523*475, closely supporting 466 already obtained value. regard <*02. ( 4)="4402*56" Â±1, 3 f\. 44576 53446 put because appear satisfy dx' x, x. x 98*46 large (3). 5. p observed. (5), e. t '(5n> = `