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Full text of "Lectures On Theoretical Physics"

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LECTURES ON 
THEORETICAL PHYSICS 



MACMILLAN AND CO., LIMITED 

LONDON BOM HAY - CALCUTTA . MADRAS 
MELUOUKNE 

THE MACMILLAN COMPANY 

MEW YORK HUSTON . CHICAGO 
DALLAS SAX FRANCISCO 

THE MACMILLAN CO. OF CANADA, LTD. 

TORONTO 



LECTUEES ON 

THEOBETICAL PHYSICS 

DELIVERED AT THE UNIVERSITY OF LEIDEN 

BY 

H. A. LORENTZ 

AUTHORISED TRANSLATION 

BY 
L SILBERSTEIN, Pir.I)., AND A. P. Jl. TKIVKLLl 



VOLUME I 

AETHER THEORIES AND AETHER MODELS 
EDITED UY II. BREMEKAMP, Pn.I). 

KINETICAL PROBLEMS 
EDITED JBY E. D. BRUINS, Pu.D., AND J. REUDLER, Pii.D. 



Osmania University HYO - i. 



MACMILLAN AND CO., LIMITED 

ST. MARTIN'S STREET, LONDON 

1927 



COPYRIGHT 



PRINTED IN GREAT BRITAIN 
BY It. & K. CLARK, LIMITED, EDINBURGH 



PREFACE TO THE ENGLISH TRANSLATION 

A SERIES of admirably clear and instructive courses of lectures, 
covering essentially almost the whole field of Theoretical Physics, 
were delivered by Professor Lorenf z at the University of Leiden. 
Seven of these courses, bearing on subjects of fundamental 
importance, were, up to 1922, edited by Lorentz's pupils and 
published in the Dutch language. During Professor Lorentz's 
visit in the United States in 1922 the plan was conceived to make 
these courses of lectures accessible to. English readers. 

After some delay necessitated by circumstances the present 
Volume I. is being issued. It contains the English version of 
two of these courses of lectures, viz. Aether Theories and Aether 
Models, edited in the Dutch language by Prof. H. Bremekamp, 
and Kinetical Problems, edited by Dr. E. D. Bruins and Dr. 
J. Heudler. To preserve the peculiar charm of Lorentz's own 
style and exposition the translation of these lectures has been 
made as literal as was compatible with the nature of the English 
language. Only a very few changes or explanatory additions 
were made in the text. The latter are placed in square brackets. 

Volumes II. and III., now in preparation, will contain the 
remaining five lecture courses, namely, Thermodynamics, Entropy 
and Probability, Theory of Radiation, Theory of Quanta, and The 
Principle of Relativity for Uniform Translations (Special Theory 
of Relativity). 

Recently, 1925, one more course of lectures, on Maxwell's 
Theory, was brought out at Leiden, and arrangements will be 
made to include its English version into this publication. 

L. S. 

A. P. H. T. 
ROCHESTER, N.Y., 
July 1926. 



CONTENTS 

AETHER THEORIES AND AETHER MODELS 
I. ABERRATION OF LIGHT 

PACK 

1. Stokes 1 Theory : The Earth drags the Surrounding Aether . . 3 

2. Velocity Potential in an Incompressible Aether ... 6 

3. Planck's Theory. Compressible Aether .... 7 

4. Fresnel's Theory. Fixed Aether ..... 12 

5. Dragging Coefficient ....... 13 

6. Theory of Aberration ....... 14 

7. Michelson's Experiment . . . . . . .19 

8. Contraction in the Direction of Motion ..... 23 

II. MECHANICAL AETHER THEORIES 

9. Maxwell's Equations ....... 24 

10. The Magnetic Force as Velocity. Neumann's Theory of Light . 26 

11. The Electric Force as Velocity. Fresnel's Theory of Light . . 29 

12. Theory of Elasticity. McCullagh's Aether .... 30 

13. Quasi-rigid Aether 31 

14. Quasi-labile Aether ....... 35 

15. Graetz's Theory ........ 36 

16. The Charging of a Conductor according to the Elastic Aether 

Theories ......... 37 

III. KELVIN'S MODEL OF THE AETHER 

17. Kelvin's Model of the Quasi-rigid Aether .... 40 

18. Solid Gyrostat 44 

19. Liquid Gyrostat ........ 48 

20. Liquid in Turbulent Motion as Aether Model .... 52 

IV. ATTRACTION AND REPULSION OF PULSATING SPHERES 

21. Nature of the Problem 58 

22. A Single Moving Sphere ....... 59 

23. A Sphere at Rest in a Liquid with a given Motion ... 60 

24. n Spheres ......... 61 

25. Two Spheres ........ 62 

26. Treatment of the Problem by means of Lagrange's Equations . 65 

27. Pearson's Theory ........ 67 

SOURCES AND REFERENCES ...... 70 

vii 



viii CONTENTS 



KINETICAL PROBLEMS 

PAOR 

INTRODUCTION ......... 75 

CHAPTER I 
INNER FRICTION AND SLIDING, TREATED HYDRODYNAMICALLY 

1. Hydrodynamical Equations of an Incompressible Viscous Liquid . 77 

2. Effect of Sliding upon a Liquid flowing in a Tube ... 80 

3. The Dragging of a Liquid by a Moving Plate .... 81 

4. Effect of Sliding in the Case of Translational Motion of a Sphere in 

a Liquid ......... 83 

5. Liquid Motion due to an Immersed Vibrating Plate ... 87 

6. Effect of Frequency in the Case of a Sphere vibrating in a Liquid . 89 

7. The Question of Validity of Stokes' Law for Brownian Movement. 93 

8. Deduction of Einstein's Formula for the Mean Square of the 

Deviation of a Particle in Brownian Movement ... 94 



CHAPTER II 
FRICTION AND SLIDING, TREATED KINETICALLY 

9. Friction Independent of the Density of the Gas ... 98 

10. Uniformity Considerations ...... 102 

11. Kundt and Warburg's Experimental Investigations . . . 107 

CHAPTER III 

KNUDSEN'S INVESTIGATIONS ON RAREFIED GASES 

12. Flow of a Rarefied Gas through a Narrow Tube . . 1 10 

13. Knudsen'H Experimental Investigation on the Flow of a Rarefied 

Gas through a Narrow Tube . . . . . .117 

14. Flow of a Rarefied Gas through a Narrow Orifice ; Knudsen's 

Experiments ........ 120 

15. Flow of a Rarefied Gas through a Narrow Tube whose Ends have 

Different Temperatures . . . . . .122 

1(5. Mutual Repulsion of Two Plates at Different Temperatures separated 

by a Rarefied Gas J2f> 

17. Knudsen's Manometers . . . . . . .132 

18. Knudsen's Accommodation Coefficient . . . . .134 

19. Heat Conduction in a Rarefied Gas contained between Two Plates 

of Unequal Temperature . . . . . .136 

20. The Effect of Accommodation upon Heat Conduction . . 140 

21. Heat Conduction in a Gas of Greater Density contained between 

Two Plates of Unequal Temperature . . . .145 



CONTENTS ix 

PAUK 

22. Heat Conduction Coefficient Independent of Density . . .149 

23. Lasareff's Experimental Investigation . . . . .150 

CHAPTER IV 
REMARKS ON LESAOE'S THEORY OF GRAVITATION 

24. Lesage's Theory of Gravitation 151 

25. An Electromagnetic Analogue of Lesage's Theory . . . 153 

CHAPTER V 

FRICTION AND HEAT CONDUCTION IN THE PROPAGATION 
OF SOUND 

26. The Effect of Friction 156 

27. Effect of Heat Conduction 160 

CHAPTER VI 

KINETIC THEORY OF SYSTEMS OF ELECTRONS 
RICHARDSON'S INVESTIGATIONS 

28. Theoretical Introduction 163 

29. Validity of Maxwell's Distribution Law for the Free Electrons in a 

Metal 167 

30. Velocity Distribution of Thermions for Different Directions . .173 

31. The Work required to drive an Electron out of the Metal . . 175 

32. The Density of Electrons in the Metal 177 

33. Difference in Potential Energy between the Electrons inside and 

outside the Metal Atoms 179 

CHAPTER VII 
VACUUM CONTACT OF PLATES OF DIFFERENT METALS 

34. Potential Difference for Vacuum Contact . . . .182 

35. Resistance at a Vacuum Junction . .184 

36. Peltier Effect for a Vacuum Contact 185 

37. Richardson's Measurements . . . . . .186 

CHAPTER VIII 

PROBLEMS IN WHICH THE MOTION OF ELECTRONS PLAYS A PART 

38. Nichols 1 Experiment 189 

39. Kinetic Energy of the Conduction-current Electrons . . .190 

SOURCES AND REFERENCES ...... 193 



AETHER THEORIES AND AETHER MODELS 
(1901-1902) 



I 

ABERRATION OF LIGHT 

THE first question which presents itself when we try to form an 
idea of the nature of the aether is that concerning its relations 
to ponderable matter. More especially we will consider the 
question whether a moving body, such as a planet, does drag 
the surrounding aether. The theory of the aberration of light 
may give us in this respect some information. 



1. STOKES' THEORY: THE EARTH DRAGS THE SURROUNDING 

AETHER 

Stokes imagines the aether streaming in the neighbourhood 
of a planet and determines the direction in which according 
to this hypothesis a star should be seen in the following 
way. If we consider a wave-front near the planet, we can 
determine any of its successive positions by the Huygens con- 
struction, provided we take account of the velocity of the 
aether at each point of the original wave-front. Since the 
velocities at different points are different, the wave-front will 
be slightly tilted, and since the direction in which we see the 
light source is determined by the normal of the wave when it 
reaches our eye, this rotation will give us an explanation of the 
aberration. In order to find the amount of the rotation thus 
produced, we note first of all that the translational [orbital] 
velocity p of the earth is 10~ 4 times the light velocity V. Terms 
of the order (p/V) 2 will be neglected. 

Let the plane AB (Fig. 1) represent the wave-front at the 
instant t, and let our z-axis OZ be perpendicular to this plane at 
a point M. Then, if u , v , W Q be the velocity components of 

3 



AETHER THEORIES AND AETHER MODELS 



the aether at M 9 we can write for the velocity components at a 
point x, y, 2 of AB chosen in the neighbourhood of M, 

du du 



9v do 
$?+tfl' 

dw div 



A 



B 




During the time dt the point will be displaced to the new 
position 

du du \ i. / dv 



The plane drawn through all the points thus obtained is parallel 
to the new wave-front. Its equation will be 

dw dw 



This relation, in whose second term x and y stand for the old 
co-ordinates, will hold, provided the differences between these and 
the new co-ordinates are very small. The direction-cosines of 
this plane are 

fa,. dw,. 
-*-dt, -~ at, 1, 

dx ' 3.V 



i ABERRATION OF LIGHT 5 

so that the wave normal makes with the X -axis the angle + eft, 

* 2 Px 

and with the Y-axis the angle *" + * - dt. The changes of the two 

2 y 
first direction-cosines, which originally were zero, that is to say, 

the magnitudes 

dw jA , dw ,, 
*-dt and ~ at, 
dx dy 

can be taken as a measure of the rotation of the wave-front 
about the y- and the x-axis. 

The wave-front propagates itself in the time dt over a distance 
Vdt, so that the rotation, per unit length, has the components 

1 dw 1 dw 
"7 Jte' ~V~dy' 

The total rotation undergone by the wave-front until it will have 
reached our eye will, of course, be expressed by an integral. 
The preceding reasoning gives us the contribution to that integral 
for a wave- front which is normal to the adopted Z-axis. The 
deviations from that orientation remain, however, so small that 
these results can be used for all the elements of the integral. 
Thus the components of the total rotation will be 

, 1 



where the integrations are to be extended from a point at which . 
the earth's motion is still imperceptible up to the observer's 
station P. 

Now, it is actually possible to account for aberration by 
means of this rotation of the wave-front, provided that the 
motion of the aether is assumed to be irrotational. There exists 
then a velocity potential <f>, such that 

?J> ty fy 

u = ^ , v v , w = J- 
fte vy vz 



Thus also flt0/ftc = 8ti/3z, dwfiy^dvpz, and our expressions for 
the rotation components become 



6 AETHER THEORIES AND AETHER MODELS i 

It will be readily seen that this result agrees with observation, 
and with the elementary theory as well, if it be assumed that 
the velocity of the aether at the observing station P is equal to 
the translational velocity of the earth. 



2. VELOCITY POTENTIAL IN AN INCOMPRESSIBLE AETHER 

Here, however, a serious doubt arises, whether the existence 
of a velocity potential is compatible with the requirement 
that the velocity should be equal all over the surface of the 
earth (viz. equal to the earth's velocity in its annual motion), 
whereas the assumption of such a potential is indispensable for 
obtaining the correct value of the aberration. As a matter of 
fact, there is nothing uncommon about a motion for which a 
velocity potential exists. In a frictionless liquid in which there 
are, at a given instant, and therefore also at any later time, no 
vortices, no other motion, in fact, is possible. But, as was just 
mentioned, the assumption of a velocity potential for an incom- 
pressible aether cannot be reconciled with the other requirement 
that the velocity at the earth's surface should be everywhere the 
same, in size as well as in direction. In fact, in an incompressible 
aether the motion is completely determined, if it be assumed 
that a velocity potential exists and that the normal velocity 
component of the aether at the surface of the earth is every- 
where equal to that of the translation velocity of the earth ; but 
the tangential components are then found to be different. If 
the origin is at the centre of the earth, the Jf-axis in the direction 
of the lianslational motion of the earth, and if the velocity of 
this motion is p, then all requirements are satisfied (and according 
to the theory of the Laplace equation this is the only possible 
solution) by the potential 

. Cx 

9= yS' 

so that 



where C is determined by the condition that the normal com-, 
ponent of the velocity at the surface of the earth should be 
equal to the component of p in the same direction. If R be the 



i ABEKRATION OF LIGHT 7 

radius of the earth, and the angle between the direction r and 
the -3T-axis, then, since 3</?)r= -2Ck/r*, the last condition gives 



and therefore C = - 1 

Now, in order to see that the Velocity of the aether cannot be 
the same all over the surface, viz. equal to that of the earth, it 
is enough to notice that for points of the surface in a plane 
through the centre and perpendicular to OX the velocity along 
the X-axis becomes -\p, whereas at the intersection point of 
the surface with the X-axis we find the velocity p. In order to 
avoid this difficulty, one might perhaps take advantage of the 
circumstance that the velocity potential need not exist in the 
whole space around the earth, since we are concerned only with 
a limited region. This, however, would lead to very artificial 
and improbable concepts. 

3. PLANCK'S THEORY. COMPRESSIBLE AETHER 

It was shown by Planck how these difficulties can be met by 
giving up the incompressibility of the aether and by assuming 
that the aether is subject to the earth's attractive force. To 
investigate the details of the motion, it will be convenient to 
attribute to the whole universe a velocity -p, while consider- 
ing the earth to be at rest. AVe have then to investigate the 
disturbance produced by the fixed sphere in the otherwise 
uniform aether stream. Planck adjusts his hypotheses so that 
the velocity at the surface of the sphere should be small. In 
order to see how this requirement can be satisfied, it is enough 
to realise that a plane through the centre of the sphere perpen- 
dicular to the direction of the stream must be traversed in a 
given time by the same amount of aether as in the absence of 
the sphere. The quantity of aether which in the latter case has 
to stream through the circle cut out by the sphere from that 
plane, must find its way through the remaining part of the plane 
surrounding the circle, and especially through that nearest to 
the circle. This is, with a small velocity, only possible if the 
density in the neighbourhood of t|jp sphere is large. It is 
necessary, therefore, to assume that the aether [to be thus 
condensed] is attracted by the earth. It must further be assumed 



8 AETHER THEORIES AND AETHER MODELS i 

that the velocity of light in this very condensed aether is the 
same as in the aether of normal density. Planck assumes also 
that the aether while being condensed behaves like a gas. 

We will now work out in detail these ideas. Let the Z-axis 
be chosen in the direction of motion of the aether at infinity. 
Let, as before, < be the velocity potential and, therefore, the 
velocity components u=d<f>j?x, v=9$/fy, w^ty/dz. The dis- 
tance of a point (x, y, z) from the centre of the earth will be 
denoted by r, and the radius of the earth by r . Let p be the 
pressure, and k the density of the aether, and let k/p=fju 
be assumed to be constant. Further, let V be the potential of 
the attractive force, per unit mass, so that the components of 
this force, again per unit mass, are -?V/?x 9 -dVfiy, -dVfiz. 
Then V = qjr, and the force, which is radial, is, per unit mass, 
~dVfir=qlr 2 , and therefore at the surface, ?/r 2 = -g. Thus q 
is determined, and we have 

F=-^ 2 . 
r 

Let us now write down the equations of motion for a stationary 
state, in which the velocity find the density at every point have 
always the same values. First of all, the equation of continuity, 



i <\ 1 1 \ 

~ J 1=0. . . . (1) 
?z v ' 



Next, the equation of motion, 



~ ~- ~-= - ~ - 7 *, 
vx dy vz vx K vx 

where the left-hand member represents the acceleration along the 
X-axis, since, the state being stationary, ?u^t=^0. The right- 
hand member is the force per unit of mass in the same direction. 
Two similar equations will hold for the Y- and the /-directions. 
Owing to the existence of a velocity potential an integral can be 
found at once. In fact, the equations of motion can be written 



etc., whence 



i ABERRATION OF LIGHT 9 

Let us now assume that the term containing the square of the 
velocity can be neglected. Then, 



This equation shows that, to the assumed degree of approxima- 
tion, the density distribution in the aether can be taken to be 
the same as in the state of rest. Thus we can write 

1 r 2 

- log p -g = const., 

or also 

1 r 2 

- log k -g = const. 

The integration constant can be expressed in terms of the aether 
density A* at the surface of the earth, 

1 , , r 1 



instead of which we can write 

i * f } l \ 
log , -al - , 

b *o Vr V 

where a=W^o 2 - 

For r -= oo we find the limit value of k, 



In the next place, Planck determines the velocity potential 
from the equation (1), where now k is a known function of the 
co-ordinates. What is required is a solution representing a 
motion which at large distances reduces to a stream along the 
Z-axis. Let 7 be the velocity of this stream.* Then, at large 
distances, 0=72. This suggests the form <f>=zf(r). Then, 

ty =df 9 djjzdf^t = z*df 
to r dr dy r dr' dz r dr *' 



_ 
rdr r 3 dr r 2 cfr* 1 



* This was previously denoted by -p. In accordance with Planck's 
notation p will now be used for the pressure. 



10 AETHER THEOEIES AND AETHER MODELS 

?#-4fj&#.0* 
fy 2 rdr r 3 dr r 2 ' 

&<f> = Zz<tf_**<y .# 
dz* r dr r*dr r*r 

and the equation (1) becomes 

j/ A zdf d?f\ dk xzdf ?>k yzdf dk/z 2 df 

&(4- / + 25,-,)+^- ' ;/ +*' ' j+i\-j + 
\ rdr dr*/ fa r dr ty r dr dz\r dr J 

Substituting here the value of log k just found, we have 
A zdf d?f I x xzdf [yyzdf zfz*df 

4 i +Sja- a "La -y+.a /+ai ^ 
rar ar 2 (r 3 r ar r 3 r ar ^Vr dr 

or, after some reductions, 

(Pf /4 a\df a, n 

j'i + l- - a)/ - ^i/^ - 
dr 2 \r rVrfr r 3 * 7 

The solution of this equation is 



The constants a and 6 will be determined by considering the 
state of things at infinity and at the earth's surface. For r = oo 
we must have <f>*=zy, and since <f>=--~f(r), 



and b-a = y. At the surface of the earth the aether cannot 
have any velocity perpendicular to that surface, i.e. 
and since 



we have, at the earth's surface, /(r) +rf'(r) =0. Substituting the 
value of/(r) and putting r = r , we find 



- 
r \2r 

Thus a=fe,r 2 2 + a + ]e-^ (4) 

2 v ; 



i ABERRATION OF LIGHT 11 

Now for the sliding of the aether along the earth's surface. 
In order to find this, we have to calculate the Z-component of 
the velocity. From <f> = zf(r) follows 



and, at the surface of the earth, where /(r) +rf'(r) = 0, 



where 6 is again the angle between r and the Z-axis. If ro is 
the velocity in the tangential direction, then, since fty/fe = & sin 0, 
we have 

=sin0./(r), 

whence, by (3) and (4), 



r 

2r 

Our aim now is to make the coefficient of sin small in com- 
parison with 7. This will be the case if a/r is large. For then, 
in virtue of (4), the constant a will be small as compared with 6, 
and therefore, since b- a = y,b will be approximately equal to 7. 
Planck has made some numerical estimates under different 
assumptions. Let us first take p^k/p and a^=/igrr 2 as great 
as for air at C. and for g the value which holds for 
ordinary ponderable matter. Then the consequences hold for 
the air actually surrounding the earth. Thus, fl/r -800, and 
the ratio of the density at the earth's surface to that prevailing 
at a large distance is 

e a/r 0== g800 

The result of the computation is therefore, practically, that our 
atmosphere can well be kept by the earth, though at the same 
time the theory shows that some residual sliding is unavoidable, 
provided that friction be disregarded. Now, with reference to 
the aether we need not choose our assumptions so as to be 
driven to such an extravagant condensation. It can easily 
be calculated how far we must go with this/^The aberra- 
tion constant is known only up to ^ per "'cent. In other 
words, the effect of a [sliding] velocity smaller than vj^ 



12 AETHER THEORIES AND AETHER MODELS i 

of the velocity of the earth cannot be detected. It is enough, 
therefore, that the velocity of the aether at the earth's surface 
should remain below this limit. For this purpose it is only 
necessary to have a/r >ll. Even so we are still left with a 
condensation e 11 [or about 60,000], The natural question arises 
whether such a strong condensation could not be tested in an 
independent way. Notice that the condensation around the sun 
and similar bodies will be considerably greater, since the attrac- 
tion is proportional to the mass. The aether condensation due 
to the presence of the sun will also increase somewhat the con- 
densation e a/r at the earth's surface. A number of questions 
suggest themselves here, but must be left unanswered. 

4. FRESNEL'S THEORY. FIXED AETHER 

We now turn to the alternative hypothesis, that of a fixed 
aether. This was already assumed by Fresnel. The aether 
must then be able to pass freely through the earth. The atoms 
themselves may well be impenetrable, provided, however, that 
they are assumed to occupy but a small fraction of the total 
volume. We may also assume that the atoms are not impene- 
trable, following a line of thought according to which the atoms 
or their constituents are but special modifications [singularities] 
of the aether. 

In order to explain aberration we again apply Huygens' 
principle. Let us take the simple case in which the position of a 
star is being determined by means of a primitive sighting appar- 
atus (without lenses), and let the presence of the atmosphere be 

disregarded. The explanation is 
^ *L then the same as on the emission 

theory. The propagation of the 

* -- wave -front which at a certain 

F ICU 2. instant reaches the aperture AB 

(Fig. 2) is unaffected by the 

motion of AB. A simple construction gives us that portion 
of the screen CD which receives the aether disturbance, 
and it is manifest that in this way one falls back to the 
older explanation. But this reasoning cannot be applied when 
we are concerned with refracting surfaces. In this direction 
many experiments on aberration were made. As a typical case 




i ABERKATION OF LIGHT 13 

we can consider the famous experiment of Arago. If ab (Fig. 3) 

is the direction in which, with the aberration, a star would be 

seen directly, then cd, the direction 

in which the star is seen through a 

prism, will coincide with that of the 

refracted ray belonging to ba as 

incident ray. Another experiment ^ F IG . 3. 

[Boscovich - Airy] proves that, in 

observing a star, a telescope filled with water has to be set in 

the same direction as an ordinary telescope. To sum up, these 

experiments show that all refraction phenomena arc the same as 

if there were no aberration. 

5. DRAGGING COEFFICIENT 

In order to explain Arago's experiment Fresnel introduced 
the dragging-hypothcsis, which amounts to this : 

Let W be the propagation velocity of light in a given medium, 
when this is at rest. Then, if the medium is moving with a 
velocity p in the direction of light propagation, the velocity of 
light relatively to the aether is, according to Fresnel, not W +p 
but W +kp, where k is a fraction. In other words, the ponder- 
able medium behaves as if it 
dragged the light with a velocity 
* n which is only a certain fraction 

of its own velocity. The co- 
efficient k must then be 

fc- 1--1, 

w 2 

where n is the refractive index 

of the medium when at rest. 

Fj ' 4 ' For w = l we have fc = 0, as it 

should be ; for when n tends to unity, the medium becomes 
indiscernible from the aether. 

This expression for the coefficient k can be easily found by 
the following reasoning. Let us suppose that the position of a 
celestial object is being determined by means of a sighting 
apparatus consisting of a screen with an aperture AB (Fig. 4), 
which is fixed in the aether, and of a second screen with an aper- 
ture CJD, behind which is placed some ponderable medium, as 



14 AETHER THEORIES AND AETHER MODELS 

e.g. glass, and finally of a third screen which receives the light. 
Since the refraction is such as if the apparent direction of the 
ray were the true one, the segment EF of the third screen which 
receives the light will be found by putting 



sin (ACH). 

This is the result of the experiment. Now, let V Q be the velocity 
of light in the aether, v l the light velocity in glass (at rest), and 
p the velocity of the earth, and let us consider the case in which 
the incident light is perpendicular to the first screen. Then, 
since the deviation from the normal remains small, 

sin (ACH) == tan (ACH) ^pjv Qt 

and therefore, ^ECG^p/nv^ and EG = lp/nv , if l = CG. The 
segment thus determined differs from that which we would find 
if, starting with the wave-front CD, we applied Huygens' prin- 
ciple and used in it v l as light velocity ; for then we should find 
a segment E'F' such that 

E'GiCG-piv^pivJn, 

and therefore, E'G = lnplv Q . To reconcile this with the experi- 
mental result we have to assume that within the time J/t^ the 
glass has dragged the light over a distance 

E'E = E'G - EG = Inpfa - lp/nv Q , 
that is to say, with the velocity 



6. THEORY or ABERRATION 

If this dragging coefficient is assumed, it can be proved that 
all the phenomena of refraction, etc., are such as if there were no 
aberration. For this purpose we use the artifice of imparting to 
the whole system a velocity equal and opposite to that of the 
earth. The earth is then at rest, while the aether has everywhere 
the same velocity. This proof can be given for the more general 
case of any motion of the aether, provided it has a velocity 
potential. Our first business is to find out how the wave-fronts 
and the light-rays are to be determined. The elementary wave- 



i ABERRATION OF LIGHT 15 

front spreading out from a point would, after a time t, in a 
stationary aether, be a sphere of radius vrf. Now, if p be the 
velocity of the aether at 0, and if it varies continuously from 
point to point, then, neglecting infinitesimal terms of the second 
order, it can be shown that this sphere is simply displaced as a 
whole over the distance pt. The successive wave-fronts are the 
envelopes of the spheres thus determined, every time to be con- 
structed around the points of the preceding wave-front as centres, 
and taking account of the aether velocity at those points. 

To find the wave-fronts in a ponderable medium, let us 
first consider the case in which the aether is at rest, while the 
ponderable medium has the velocity - p. The elementary wave- 
front is then, after a time t, a sphere of radius vj, which is dis- 
placed as a whole over a distance -pt(\ - 1/n 2 ). If we now give 
to the whole system the velocity p, the elementary wave-front 
will be displaced relatively to the 
ponderable medium, thus brought to 
rest, over the distance O^^ ^X^i* 

_1 

n ' -TO/ 

where /c = l/n 2 . This coefficient can 

be said to determine to what extent 

the light has been dragged by the aether. Let now (Fig. 5) 
be a point of a wave -front. The elementary wave -front 
emanating from is a sphere with 0' as centre, where 00' = icpt, 
and of radius vtf. Let A be the point of contact of this sphere 
with the envelope which is the new wave-front. Then OA will 
be an element of the light-ray through 0, and we find for the 
propagation velocity along this ray, relatively to the earth, 
w^OAjt. Thus, if be the angle OVA, we have 

v^ =w 2 - 2i<wp cos + K 2 p 2 , 
whence, neglecting higher powers of p 9 

30S 2 0-1) ... (5) 

w V} [ Vi 20J 2 J * 

These formulae hold for any continuous space-distribution of the 
velocity p. 




16 AETHEE THEORIES AND AETHER MODELS i 

Having thus found the velocity along the ray, we can readily 
determine the light path relatively to the earth. For this pur- 
pose we make use of a theorem according to which light follows 
that path to which corresponds the shortest time, a theorem 
which in the present case can also be easily deduced from 
Huygens' principle. In fact, let AB be the light-ray determined 
by a Huygens construction, and let us consider any other path 
between A and B. Both paths cut the successive wave-fronts. 
The intersection points of the latter with the actual light-ray 
may be called corresponding points. Let S and S' be two 
successive positions of a wave-front separated by a very short 
time-interval, and let P', Q', etc., be points of S' corresponding 
to P, Q, etc., chosen arbitrarily on S. Then the time of passage 
along PP', QQ', etc., always with the ray velocity belonging to 
these lines, will be the same. The time required for covering 
any other path drawn from S to S', again with the ray velocity 
belonging to it, that is to say, a path laid through non-corre- 
sponding points, will be longer. This appears from Fig. 6, 
in which A is a point of the first wave-front and A'D lies in 

the second wave-front, A' being the point 
AL jw in which the latter wave-front touches the 

elementary wave A'C circumscribed around 




/A* A, so that A' and A are corresponding 
FIG 6 points. Since the light time for A A' is as 

long as that for AC, that for AD is evi- 
dently longer. Let us now imagine between the points A and 
B the whole series of successive positions of the wave-front. 
Then, remembering that the light-ray is the locus of corre- 
sponding points, while this cannot be said of any other path 
from A to B 9 it will* become manifest that the time along 
the light-ray is indeed the shortest among all paths leading 
from A to B. 

This being proved, the light path can be determined in the 
following way. The time required for covering an arbitrary 

path between A and B is I dsjw, the integral to be extended 

JA 

over the path. Now, this expression has to be a minimum for 
the actual light path. 

Let us first take the case of a non-homogeneous (but isotropic) 
ponderable medium, without surfaces of discontinuity, however. 



i ABERRATION OF LIGHT 17 

Then tc , upon which w depends, is variable, since n as well as v l 
are variable. Further, w depends also on p and 0, which in their 
turn depend on the state of motion of the aether, of which we 
have only assumed that it has a velocity-potential <. 

Thus, taking account of (5), and neglecting terms of the order 

2 , we find for our integral 



I* 1 - r f\ \ I 

- -*- cos ) = / - 

~~ 'A v i 'A nZv i* cte" ' 

Now, nv^VQ being constant throughout the space, the last 
integral is (<j3-<ki)> an d therefore independent of the path 



between A and B. The minimum property is thus influenced 
only by the first integral on the right hand, and since this con- 
tains no trace of motion, the light path will be the same as if 
there were no motion of the aether. Notice that this proof is 
based, first, on the assumption of Fresnel's dragging coefficient 
and, second, upon the existence of a velocity-potential. 

It follows from the Huygens construction that the minimum 
property holds also for the case in which the light passes from 
one to another medium. Consequently, the light path can in 
this case be found by the same reasoning as before. In fact, 
considering any path from A to B which cuts the refracting 
surface at C, and putting l/w 2 ^ 2 ^/*, we have 



* 



In order that this should be a minimum, it is enough to make 
the first term a minimum. Consequently, the light-rays in 
reference to the earth obey the ordinary laws of refraction. 
We have still to consider the relation between the light-ray 
VOL. J c 




18 AETHER THEORIES AND AETHER MODELS i 

and the wave-front. For these are no longer perpendicular to each 
other. Let (Fig. 7) be a point of a wave-front, let 00'=jM, 
and let P be the corresponding point of a successive wave-front, 

so that OP=wt will be an 
element of the light-ray, and 
O'P v^t. If now the wave- 
front, and therefore the 
wave-normal (O'P), at every 
FIG. 7. / * point be given, then the 

direction of the light-ray 
can be found by combining the velocity v l along the normal 
with the velocity *p. Vice versa, if the ray be given, the direction 
of the wave-normal will be found by combining the velocity w 
along the ray with the velocity - up. In the latter construction, 
and up to terms of the order j? 2 /v 2 , the velocity w can be replaced 
by v l taken in the same direction. 

Let us still consider a luminous point L in a homo- 
geneous medium, and let us determine the wave-fronts by the 
indicated construction. The light-rays being straight lines 
diverging from //, it is required to find the direction of the 
wave-normal at any point A. If x, y, z be the co-ordinates of 
A, with L as origin, the components of the ray velocity v { at 
the point A are 

x y z 
/i> /i> /i< 

and these must be compounded with -icp, the velocity of 
dragging. The components of the latter are - /tf^/rte, - /e?)0/rty, 
- icdfyfiz. Thus the components of the resultant will be 

"(ty - *<), ^(vf - K</>), ^(f V - *<), 

and since this resultant has the direction of the wave-normal, 
the equation of the wave-front becomes 

Vjr-K(f)= const. 

In the neighbourhood of L the potential $ can be considered as 
a linear function of the co-ordinates, so that the last equation 
assumes the form 

ty - K(OX + by + cz) = const. 



i ABERRATION OF LIGHT 19 

As we already know, the wave-fronts in the immediate 
neighbourhood of L are spheres whose centres lie at a certain 
distance from L. It is manifest that the equation agrees with 
this, provided terms with p 2 are neglected, as in fact they were 
in deducing it. Again, waves converging towards a point have 
the same form as would have expanding ones, if the velocities 
in the aether were everywhere reversed. 

In order to apply this reasoning to the determination of the 
direction in which we see a star through a telescope, let us 
consider a wave-front arriving from a star, and let us derive 
from the given wave-normal the direction of the light-ray. It 
is these relative light-rays which we observe in our experiments. 
In the free aether, far away from the earth, the direction of the 
relative ray is found by combining the velocity V Q along the 
wave-normal with /cp, where # 1. In this manner we find, 
obviously, the same direction as according to the elementary 
theory. The further progress of these relative rays can now be 
followed up by the ordinary laws of refraction, etc. And we 
have now to orient the telescope so that the rays thus treated 
should converge upon the intersection point of the cross-wires. 
In this way the theory accounts for all the experimental facts. 

Also the diffraction and interference phenomena are such as 
if the earth were at rest and as if we had to do with the velocity 
v v If we have, for instance, two paths between A and B, the 
interference will depend only on the difference of the light times 
along them, and this difference can be determined by using for 
each path element the velocity v r For this amounts only to 
omitting, for the two paths, the term /*(</> j - (f> ), which is the 
same for both. Here again it is only necessary to assume that 
there exists a velocity-potential, but not that the earth is pene- 
trable for the aether. The latter, however, must be assumed for 
all transparent media in order to have a reasonable explanation 
of the dragging coefficient. The simplest, after all, is Fresnel's 
theory. 

7. MICHELSON'S EXPERIMENT 

We have thus far neglected all terms having p 2 /v 2 as factor. 
See formula (.5). 

In certain interference experiments, however, the accuracy 
can be pushed so far that these second-order terms have to be 



20 AETHER THEORIES AND AETHER MODELS i 

taken into account. Such is the famous experiment of Michelson, 
already suggested by Maxwell. Let A and B be two points fixed 
on the earth, and let the latter move with the velocity p along 
AB. What is the time taken by light for a complete to-and-fro 
passage ? This will depend on whether the aether shares in the 
earth's motion or not. In the former case the time in question 
will be independent of the velocity p, but in the latter case the 
velocity of light relative to the earth will be v-p in one, and 
v+p in the opposite direction, and therefore the time required 
for a complete to-and-fro passage [up to fourth-order terms], 



v-p v+p v 

Since Michelson determines this time by means of an inter- 
ference phenomenon, the influence of the last term will still be 
perceptible, though it amounts only to a fraction of the vibration 
period. Let us see how large I must be in order to give an 
observable effect. The shift of the interference fringes due to 
a time lag of ^ of the vibration period T will be just detectible. 
. For this purpose we must have 



hence, if X be the wave-length of the light, 
or I = 3 metres. 



-v 



In Michelson's experiment a ray travers- 
rk ing to and fro a certain distance in the 

direction of the earth's motion is made to 
interfere with a ray traversing the same 
distance up and down in a perpendicular direction. Fig. 8 
gives a schematic representation of the apparatus. This consists 
of two fixed mirrors B and B' and a dividing glass plate A, 
inclined at 45 to the incident light beam. A part of this beam 
passes through the plate, is reflected at B, and after a reflection 
at the lower face of the plate enters the telescope. Another 
part of the incident light is reflected at A towards the mirror B' 
and thence through the glass plate into the telescope, where it 
interferes with the first partial beam. The experiment consists 




i ABERRATION OF LIGHT 21 

in turning the whole apparatus, including the light source and 
the telescope, by 90 and comparing the interference fringes in 
the new and the original orientation. The effect of the phase 
difference is thus doubled. Yet no displacement of the inter- 
ference fringes was observed. 

Let us consider the theory of this experiment somewhat more 
thoroughly. We have to compare the times taken by light to 
traverse two different paths be- . 
tween two points A and B (Fig. 9). 1 
We have already seen that, if only 
first-order terms are retained, the 
difference of these times is the 
same as if the earth were at rest. 
Now, to determine the effect of 
second-order terms we must take Fia 

into account that the light path 

between A and B is itself slightly changed by the motion. In 
fact, the light path could be identified with that corresponding 
to a fixed earth only when we confined ourselves to terms of 
the order p/v. Now, however, the slight change of the path 
implying second-order terms need not be negligible. But it 
can be easily shown that even with the inclusion of second - 
order effects the propagation time from A to B can be 
calculated with sufficient accuracy by extending the integral 
/dsjw, with w given by (()), along the light path which would 
correspond to a fixed earth. Let, e.g., ACB (Fig. 9) be the 
light path with the earth at rest, and ADB the actual light path 
for a moving earth. Then the integral j(Ls\w would have to 
be evaluated for the path ADB. But since the value of this 
integral is juat a minimum, it will differ from that taken along 
the path ACB only by a quantity of the second order in the 
deviations of the two paths, such as CD. And since, as 
already mentioned, this deviation is itself of the order p 2 /v 2 9 
the difference between the integrals extended over ACB and 
ADB will be of the order ^ 4 /v 4 , and can thus be neglected. 

The same holds also for the other light pencil, as indicated 
in the figure by the lines without letters. 

The propagation time from A to B, T^fds/w, can now be 
calculated as in Art. 6, developing l/w into a power series of plv 
and confining ourselves to the first three terms. 



22 AETHER THEORIES AND AETHER MODELS 

Thus, T = T! + T 2 + r 3 , where, by (6), 
fds 



' COS 



(which has no influence on the phase difference), and 



In Michelson's experiment we can put K = 1, and y is, by Fresnel's 
theory, the velocity of the earth. To the phase change due to 
the turning of the apparatus the integral r 2 contributes nothing, 
so that only r 3 has to be taken care of, where = or 180 for 
the first, and = 90 or 270 for the second of the interfering 
rays. If I be the doubly covered path, then, as we already saw, 



for the first ray, and T 3 =/? 2 //t> 3 for the second ray. Thus also 
the latter is somewhat affected by the earth's motion. This 

can also be seen in the following 
simple way. A ray impinging 
upon the glass plate at A (Fig. 10) 
returns from the mirror B and 
meets it again in A', so that 
the distance over which the plate 
was displaced in the meantime 
_ is A A' =2lpjv. Consequently the 
length of the light path is 




o. 10. 2VP + l 2 p 2 lv z = 2l(l +2> 2 /2v 2 ). ' The 

difference, as compared with a 

fixed earth, is therefore lp z /v 2 , and the difference of -the corre- 
sponding time, J^ 2 /v 3 , which is the value just found for r 3 . 

As Michelson overlooked the influence of the earth's 
motion upon the light time along the path ABA', in his first 
experiment he estimated the theoretical effect twice too high. 
In this case the distance I was 1-2 metre, so that a shift of ^ of 
a fringe width was expected. The corrected theory gave only 
- 2 V of a fringe width, and this was below the threshold of reliable 
observability. Michelson, therefore, repeated the experiment 



i ABERRATION OF LIGHT 23 

with the modification that each of the two beams passed several 
times between the mirrors. In this manner the light path was 
increased to 22 metres and should have given a displacement 
of , 4 rt of a fringe width (an estimate in which account was also 
taken of the motion of the solar system). But even now the 
result of the experiment was negative in contradiction to Fresnel's 
theory. It will be kept in mind tlmt the validity of this theory 
is not limited to the case of an aether at rest as a whole, but 
extends also to types of motion of the aether for which there 
exists a velocity-potential. 

This experiment would also clash with any theory which 
attributes to the relative velocity of the aether and the earth a 
Vcilue not sensibly smaller than the translation velocity of the 
earth. In what follows, however, we shall have in mind only 
Fresnel's theory. 

8. CONTRACTION IN THE DIRECTION OF MOTION 

We can explain the negative result of Michelson's experiment 
by assuming that the length of the arms of the apparatus is 
changed by turning it through a right angle. -This change can 
be assumed to be just such as to give to r 3 the same value for 
both rays. This calls for a contraction of the path in the 
direction of motion, as compared with the perpendicular path, 
such that the corresponding light time should be shortened by 
j) 2 //v 3 . The path then has to be shortened by Ip 2 jv 2 , and there- 

v 2 
fore I by \--l. This dependence of the dimensions upon the 

orientation with respect to the earth's motion is not as strange 
as it might seem at first. In fact, the dimensions are deter- 
mined by molecular forces, and since these are transmitted 
through the aether, it would rather be surprising if its state of 
motion had no influence upon the dimensions of bodies. The 
nature of the molecular forces is not known to us. Yet, if we 
suppose that they are transmitted through the aether in the 
same way as electric forces, we can develop the theory of this 
contraction, and we then find for its amount just what is required 
for the explanation of the nil-effect of Michelson's experiment. 
This contraction would amount for the diameter of the earth to 
6-5 cm., and for a metre rod to o-J-g- of a micron. 



II 



MECHANICAL AETHER THEORIES 

9. MAXWELL'S EQUATIONS 

WE will now consider some theories of the nature of the aether. 
Such a theory must, in the first place, explain the electromagnetic 
phenomena. We shall, therefore, begin with an interpretation 
of Maxwell's equations. Here one can put different require- 
ments. One can content himself with a theory accounting for 
the phenomena in isotropic and homogeneous media, or try to 
include also the anisotropic and heterogeneous media, and so on. 
To begin with, we shall exclude the conductors only, and shall 
thus consider anisotropic as well as non-homogeneous dielectrics. 
The latter will enable us to treat the boundary conditions, and 
therefore also such phenomena as reflection and refraction. 

The magnitudes appearing in Maxwell's equations are : the 
e'ectric force E, the dielectric displacement D, the magnetic 
force H, and the magnetic induction B. The equations are 

rotH = -D, or 



"a*" 



i ao, 

c ~dt 

\W H 
c dt 

1W, 
c dt 



(7) 



. (8) 



and 



--]*. 

24 



n MECHANICAL AETHEE THEORIES 25 

9 ~~ C tit 

' <"> 

1 dB 2 

~c dt e 

divB = or- x + --+ -*~0. . . (10) 

' ex uy vz 

Further, we assume linear relations between the components of 
D and E and between those of B and H, 



D y 



B x 
By 

J5, 



where 

12 =e 21 23 = 82 e 31 = 13 5 Ml2 ^Mai* ^23 = /*32 ^31 

For isotropic bodies these coefficients are all zero, and 



We note that the usual kinematics of continuous media gives 

1 

equations of the same form as rot H== D. In fact, if every 

c 

point of a medium is displaced through an infinitesimal distance, 
of which the components f , 77, f along three axes can be considered, 
within a small region, as linear functions of the co-ordinates 
or, y, z, then the complete change in the neighbourhood of a 
point can be represented as consisting of a displacement, a 
rotation, and three dilatations or contractions in three mutually 
perpendicular directions. To see this, it is enough to remember 
that a sphere is transformed into an ellipsoid, whose conjugated 
diameters correspond to orthogonal pairs of diameters of the 
sphere. This holds also in particular for the principal axes of 
the ellipsoid. We imagine the sphere to be first displaced so as 
to bring its centre into coincidence with that of the ellipsoid, 
then find those diameters of the sphere which coincide with the 
axes of the ellipsoid, bring them by a rotation into their actual 
position, and finally give to these diameters by a dilatation or 



26 AETHER THEORIES AND AETHER MODELS n 

contraction the actual length of the axes of the ellipsoid. The 
components of the rotation are 



In a continued displacement we can compare the state at the 
time t with that at t +dt, and the last formulae give us then the 
connection of the components of the angular with those of the 
displacement velocity. 

We may now try to interpret one group of Maxwell's 
equations by taking the components of the magnetic force to 
be proportional to the displacements in the aether, or assuming 
that the essence of that force consists in such displacements. 
In symbols, let us put 



nu_ *'*Sr V-L'u ~ (]ju z ^ 

Then * = 2cmp,-;-!'- = 2cm, * = 2cmr. 



Thus, if the observed magnetic force is in its essence a displace- 
ment of the aether particles, proportional to it and having the 
same direction, the nature of the dielectric displacement current 
consists in rotations which are the result of the aether displace- 
ment. But we are then checked by the difficulty that there is 
in such a picture no place for a constant dielectric displace- 
ment, such as occurs always in electrostatics. Moreover, in 
a constant electric field without magnetic force the dis- 
placements f, 77, f vanish, so that there is nothing to 
distinguish such a state of the medium from one in which 
an electric field is absent. 



10. THE MAGNETIC FORCE AS VELOCITY. NEUMANN'S 
THEORY OF LIGHT 

We can avoid this difficulty by identifying with the aether 
rotations not the displacement current but the dielectric displace- 
ment itself. Then our equations have to be made to agree with 



ii MECHANICAL AETHER THEORIES 27 

Consequently, we have to put 

H f =m 9 H v =m 9 H 9 = mJ* 9 . . . (12) 

D x = 2cmp, D y = 2cmq, D 2 = 2cmr. . . (13) 

Thus, wherever there is a magnetic force, we must imagine 
an aether velocity in the direction of this force and proportional 
to it, and we have to look for the dielectric displacement 
in the rotation due to or associated with that velocity. In a 
permanent magnetic field, as e.g. around a steel magnet, we have 
thus to imagine the aether streaming along the lines of force. 

We may also notice that in such a case of continual motion 
the displacements f , 77, f would not remain infinitesimal. 

It is important to keep in mind that the interpretation here 
given implies necessarily that the coefficient m has also in a 
non-homogeneous medium throughout the same value. Other- 
wise the values (12) and (18) would not satisfy the equations (7). 
The expression 

m z D//,, 

?// fa 

for instance, would no longer have the value Snflpfit, since to 
this the terms 

?w ?) Pm ?TJ 

dy fit dz dt 

would have to be added. This difficulty would assert itself 
especially at such places where the properties of the medium, 
and therefore also the value of m, vary rapidly from point to 
point, as at the boundary of two media. This theory, therefore, 
implies that the coefficient of proportionality between the velocity 
of the aether particles and the magnetic force is always and 
everywhere the same. 

We have still to explain the second group, (9), of equations. 
Before doing so we have to consider the energy relations. The 
magnetic energy is a quadratic function of the magnetic force, 
and has thus to be interpreted as kinetic energy, while the 
electric energy will become potential energy. The kinetic energy 
per unit volume, with p as density, is 



28 AETHER THEORIES AND AETHER MODELS n 

and the magnetic energy per unit volume of an isotropic medium, 
with = 1 , 



Thus we have to assume p=m 2 , so that also the density will be 
everywhere the same. But the case is different if p is not equal 
to 1. In such media the magnetic energy per unit volume is 



and therefore p=/xw 2 , i.e. the density must be proportional to 
the permeability. According to this theory the characteristic 
feature of iron, for instance, is a large aether density within it. 
For anisotropic media the relat'ons are somewhat more compli- 
cated, inasmuch as for these the magnetic energy per unit 
volume is 



Now, if we put here II x = w^ fit, etc., then we do not obtain an 
expression representing the squared velocity multiplied by a 
certain factor. By a proper choice of the co-ordinate system the 
last expression can be transformed into 



and this can be interpreted as kinetic energy per unit volume, if 
it be assumed that the aether behaves as if it had different mass 
densities for motions in different directions. 

Let us still consider the electric energy which has to be 
interpreted as potential energy. This is, per unit volume of an 
isotropic medium, 

IDE -fli - 



The aether has thus to be attributed the property that its 
potential energy is proportional to the square of the rotation 
of its particles. The question how we have to imagine such an 
aether will be taken up in the sequel. 

This theory resembles the light theory of Neumann, who also 
assumed that the aether density is the same in all media, and 
thence deduced that in polarised light the oscillations are in the 



ii MECHANICAL AETHER THEORIES 29 

plane of polarisation. This means, with the present interpreta- 
tion, that the magnetic force is contained in the plane of polarisa- 
tion, as in fact has to be assumed in the electromagnetic theory 
of light. 

11. THE ELECTRIC FORCE AS VELOCITY. FRESNEL'S 
THEORY OF LIGHT 

As an alternative we might have interpreted kinematically 
the equations 

* _ .'/ P+P 

*\ ~~ ^ ~~~ *" A 9 v/l/V. 

dy fa c ot 

This is entirely analogous to what precedes, the magnetic induc- 
tion being now represented by the rotation of the aether particles 
and the electric force by their velocity. The formulae now 
become 

Jx ~ m dt 9 '"" dt' '* fa ., . (H) 

B x --= - 2cm'p, By = - 2cm' q, B z = - 2cm'r 

where m' is again a constant. There is something strange in 
having to consider the electric force as a velocity. For an 
ordinary conductor carrying a constant charge we would now 
have a permanent, outward or inward, stream of aether. In 
the previous theory the continuous streaming of the aether 
near a magnet was at least circuitous, which state of things 
it was easy to imagine to last invariably for any length of time. 

The electric energy must now be correlated with the kinetic 
one. Its amount per unit volume, which is iel? 2 , has to be 
made equal to 



so that p = w' 2 . Since w' is constant and e has for different 
dielectrics widely differing values, the density of the aether which 
is proportional to the specific inductive capacity must have very 
different values in different substances. As the previous theory 
approached that of Neumann, the present one resembles Fresnel's 
theory. In the latter it is assumed that the displacements of 
the aether particles are perpendicular to the plane of polarisation. 
This means, with the present interpretation, that such also is the 



30 AETHEE THEORIES AND AETHER MODELS n 

orientation of the electric force. Thus on either interpretation 
we remain in harmony with what is generally assumed about 
the light vibrations in the electromagnetic theory. With regard 
to this coincidence of the model with Fresnel's theory, we may 
still notice that to different specific inductive capacities corre- 
spond different propagation velocities of the electromagnetic 
disturbances, while Fresnel looked for the explanation of the 
different propagation velocities of light in the different densities 
of the aether within different media, and made this density 
proportional to the square of the refractive index, which amounts 
exactly to what we have assumed. With the interpretation 
treated in this section the electrically anisotropic media offer the 
same difficulties as did the magnetically anisotropic bodies in the 
previous one. 

For the interpretation of the second sot of equations, (7), we 
consider the magnetic energy which has to be identified with the 
potential energy of the aether. It is a quadratic function of the 
rotation components p, q, r. Thus, as in the previous theory, we 
have to imagine a mechanism whose potential energy is just such 
a function. 

12. THEORY OF ELASTICITY. MCCULLAGH'S AETHER 

Tn an ordinary elastic body the relations arc entirely 
different. Let , 77, be again the displacement components 
(functions of the co-ordinates ,r, y, z). Then we have, in addition 
to the displacement and rotation, the dilatations determined by 



, 

Cx " J ?y 
and the shears expressed by 



The normal tension components [stresses] are determined by 

X x = 2K{x x + 0(x x + y, + z,)}, l r ?/ - 2/1 ! ?/?/ + 6(x x + y v + z,)} , 
Z z = 2 K \z z + 6(x x + y !f + z z )}, 

and the tangential components by 

. (17) 



ii MECHANICAL AETHER THEORIES 31 

where K and are the coefficients introduced by Kirchhoff. 
The energy per unit volume is 



K(x x z + y* + 2/) + K6(x x +y y + z z ) a + 1 K(x* + y* + */). (18) 

Now, for the aether this must be a quadratic function of the 
components of rotation, 



The next task would be (and such in fact is always the posi- 
tion in the older light theories) to make such assumptions as to 
give the energy that form. 

The most radical means for obtaining this result is to assume 
for the aether the validity of a very peculiar elasticity theory, such 
that the potential energy should in every case be a quadratic 
function of the angles of rotation. This is McCullaglis aether. 
Let us see how this can be dom*. Elasticity cannot exist if there 
are no forces tending to bring back the displaced particles to 
their original position (state of equilibrium). What we are con- 
cerned with is the potential energy opposed to these forces. If 
that energy is to depend only upon the rotations, then the system 
of those counteracting forces has to consist only of couples. 
The components of the rotation being p, q, r, we have to assume 
for the components of the moment of the restoring couple, 
per unit volume, 

M x = - (a n p + a 12 7 + a 13 r), M v = - (a 21 p + a 22 ? + a 23 r), 

. . . ( 19) 



where the a's are constants satisfying the conditions <i l2 =a 21 , 
a 23 = a 32> a 3i = i3- These formulae hold also for anisotropic 
bodies. The peculiar feature of the case in hand is that this 
couple of forces should not be produced by the neighbouring 
aether particles. For such a couple would depend only on the 
relative rotation. 

13. QUASI-RIGID AETHER 

To explain these couples we may introduce a second medium 
which remains in its place, not sharing in the rotation of the first 
medium, and which exerts upon the latter those restitutive forces. 



32 AETHER THEORIES AND AETHER MODELS n 

In the first medium we must then have also some inner forces 
which do not all vanish. To see this let us recall how in the 
ordinary cases the relations Y z = Z y , etc., are deduced. Let us 
consider a parallelepipedon dxdydz and 
find the couple which produces a rotation 
about the Jf-axis (Fig. 11). The two forces 
parallel to the Z-axis, acting 
JA upon the faces dxdz, give the 

!' couple Zydxdzdy, and those 

parallel to the Y-axis applied to 

the faces dxdy yield - Y^xdydz ; 




Fia. n. in all, (Z u - Y 2 )dxdydz. Now, this 

couple together with M^dxdydz 

must vanish, otherwise it would give an infinite angular 
acceleration, since it is of the third order, while the moment 
of inertia of the parallelepipedon is infinitely small of the fifth 
order. Therefore, for our aether, 

Z u - Y z - M^ 

Thus there must exist tangential stresses. Similarly, of course, 
we find 

X Z -Z X = -My, 



Moreover, we will make our assumptions as simple as possible. 
To this end we make the normal stress components zero, 
X x = Y y = Z 2 = Q, and take for the tangential stresses 



- X Z = Zjf = *(My, 



In order to see that these assumptions are in agreement with 
the adopted energy expression, it is enough to calculate the 
energy of the medium within a given surface for a stationary 
state. Let da- be an element of this surface whose outward 
normal makes with the axes the angles x, //, i/. Let us imagine 
that the displacements , 77, f are produced by some external 
forces upon the surface elements, and that these forces as well 
as the displacements and stresses gradually mount to their final 
values. 



u MECHANICAL AETHER THEORIES 33 

The force acting upon the element do- in the direction of the 
X-axis is 

(X x cos A + X y cos p + Xg cos v)da, 

and similarly for the F- and Z-components. Since all the magni- 
tudes increase proportionaDy to each other, we find for the total 
work upon the element da- 

I {(X X cos A + X y cos p +X Z cos v) +rj(Y x cos A + Y v cos ^ 

-f Y z cos v) + (Z X cos A + Zi if cos /JL -f Z z cos v)}da. 

A simple reasoning will show that the energy contained within 
the surface a- is given by this expression integrated over the 
surface. This integral can be transformed into the volume 
integral 



dr being the element of the enclosed volume. Thus, the energy 
per unit volume is 

0, 



or, by (11), (15), and (16), 



^+Z (20) 



+ \(Z V + Y,)y, + (Z t , - Y,)p+i(X t +Z t )t f + (X, - Z,) ? >. (21) 
In fact, we have 



etc., and in order to see that the remaining terms disappear, it 
is enough to remember that we are dealing with a stationary 
state. Thus, e.g., the coefficient of fin (20) becomes 

sx <>x y dx t \_ 

"" 



Now, if the energy (21) is to depend on the rotations only, 
then in the first place the terms containing the dilatations must 
VOL. i D 



34 AETHER THEORIES AND AETHER MODELS n 

vanish, i.e. Jf a ,= r i/ =Z 2 = 0, and in order to get rid of the in- 
fluence of the shears, we must have Y x +Y y =Q, Z y +Y z =Q, 
X Z + Z X = Q. Thus we fall back upon the previous values of the 
stresses. 

The energy per unit volume now becomes 



and it remains to be seen how in the two developed theories the 
coefficients a are correlated with the magnitudes appearing in 
the electromagnetic equations. In the first case (Art. 10) we 
had 



D x = %cmp, D,f = 2ciw/, D z = 2 

and the aether density was p--=m 2 . 

The potential energy must coincide with the electric energy. 
The latter is, per unit volume, 

1( U D X 2 + etc. + 2 r l2 D x D u + etc.), 
since S x = f ll D x + f l2 D u + 7 13 D 2 , etc. 

Thus the required coincidence calls for the following relations : 
a n = 4' n c 2 m 2 , a 22 = 4 / 22 c 2 m 2 , a 33 = 4' 33 c 2 m 2 , 
a = 4 / 



For isotropic media we shall simply have E = e'D, a==46 / c 2 m 2 . 
The second set of equations (9) now follows from the equa- 
tions of motion of the aether, 

ax. az. ax, w 
ib + ty + fa ~ p ~dt* etc - 

In fact, putting here 

X x =0, X v = %M lt X,*= -\M U , 

, .flM, Mf \ 9 2 | 

we have ^-^^ 

Now, 

Af.*n)Z 

etc. 



ii MECHANICAL AETHEE THEORIES 35 

Substituting this and replacing 9/5j by HJm, we find 

3tff_a?ir__l3fl ? 

dy dz c St ' 

which is the first equation of the second set (for the sub-case 

p- 1 )'. 

Similarly, in the second case, the set (7) of equations can be 
deduced from the equations of motion of the aether by con- 
sidering the magnetic energy. 

Attention to such an aether was drawn by Kelvin, who 
called it the quasi-rigid aether. With its aid he proposed to 
account for the magnetic phenomena, and considered, there- 
fore, the magnetic induction as the rotation of the aether elements, 
which coincides with our second case. 



14. QUASI-LABILE AETHER 

An explanation of the phenomena can also be arrived at 
without making about tho elasticity of the aether such uncommon 
assumptions. It will be enough to consider the case of an isotropic 
homogeneous medium. Let us substitute in the equations of 
motion 

Mk MT V MT, 



the stress components as given by the ordinary theory of elasticity, 
to wit, 



Then the result will be 



We have now to make such assumptions that the left- 
hand members of these equations should become the rotation 



36 AETHER THEORIES AND AETHER MODELS n 

components of a vector. This will be the case if we pat = - 1. 
Thus, in fact, the last equation will become 



\Sy ~ 
or, in terms of the rotations, 



which is the required form. 

With the interpretation of Art. 10 this equation is transformed 
into the first of the equations (9) for an isotropic medium, and 
with the interpretation of Art. 11 into the first of (7). 

The medium just considered is Kelvin's quasi-labile aether. 
The equilibrium of this aether is labile, the potential energy in 
the state of equilibrium not being a minimum. In fact, it is 
not difficult to see that the potential energy can under the stated 
circumstances become negative. For its expression, per unit 
volume, is 



and this, with = -1, is obviously negative provided that the 
shears vanish and x x , y y , and z g have the same sign. 

This aether is perhaps not so satisfactory as the quasi-rigid 
one, because the corresponding theory is limited to isotropic and 
homogeneous media, so that also the case of boundary surfaces 
has to be left out of account. 



15. GRAETZ'S THEORY 

The last mechanical aether theory to be still considered is 
due to Graetz. In this the second set of equations is obtained 
in the same way as with the quasi-labile aether. In fact, what 
in the last case was achieved by putting 6 = - 1 in the equation 
of motion of an ordinary elastic medium, 



with P written for + + and A for + 



% [a 



ii MECHANICAL AETHER THEORIES 37 

Graetz obtains by adding to the right-hand member a term 
- 2K(l + QfiPfix, which converts it into 



His theory then amounts to retaining for the free aether the 
ordinary elastic equations, so that in a vacuum Maxwell's equa- 
tions do not hold. After all, since the free aether cannot be 
experimented with, we shall never be able to find out whether 
these equations do hold for it or not. For a ponderable medium 
Graetz assumes that the aether particles are acted upon not 
only by the surrounding aether but also by the ponderable 
substance. The force due to the latter gives then the term 

-2K(l+0)dP/dx. 

Of this Graetz gives also some account, inasmuch as he deduces 
the said force from a pressure exerted by the ponderable matter 
upon the aether and represented by p = 2K(l + 0)P. 

16. THE CHARGING OF A CONDUCTOR ACCORDING TO 
THE ELASTIC AETHER THEORIES 

We have just seen how it is possible, by ascribing to the 
aether various properties, to account for Maxwell's equations. 
We must come back, however, to a certain difficulty which 
appears in all mechanical aether theories and which presents 
itself when we consider a charged conductor. If we look 
upon the electric force as the manifestation of a velocity in the 
aether, then, as was already mentioned, we must imagine a 
continuous aether stream towards or from the conductor, though 
we perceive in the latter no change whatever. 

How have we to picture to ourselves a charged conductor on 
the theory in which the magnetic force is represented by an 
aether stream ? The dielectric displacement is then a rotation 
about the lines of force in the sense corresponding to that of the 
lines. The components of this rotation are p = %(dldy-drjldz), 
etc., whence we see that the distribution of the rotation is always 
solenoidal. Let through every point of space a vector be drawn, 
indicating the direction of the rotation. Then the system of 
curves to which these vectors are tangential will be the so-called 



38 AETHER THEORIES AND AETHER MODELS n 

vortex lines, and of these, owing to the solenoidal distribution, 
we can also construct vortex tubes. 

Let us now consider a spherical conductor. During the 
process of charging, electricity is being communicated to it 
through a wire. This means that rotations are being produced 
in the aether, and therefore angular velocities exist while the 
conductor is being charged. Let us, then, consider the vortex 
lines in the whole space surrounding the sphere. In virtue of 
the solenoidal distribution these lines must either be re-entrant 
or extend from infinity to infinity. To each of these vortex lines 
must correspond another within the wire. One would now 
have to imagine that the aether fibres inside the wire are twisted 
and that this twist is propagated through the conductor and 
thence along the lines of force in the medium. The rotation in 
the latter is opposed by the elasticity of the medium giving rise 
to couples which, in absence of the rotating force, would at once 
untwist the system, that is to say, discharge the conductor. 
Now, the difficulty consists in finding out what happens when 
the wire is removed. For the state of affairs which actually 
takes place cannot exist in the picture just given. 

In order to see this, we consider an arbitrary surface or, 
whose normal at a point x, y, z has the direction angles a, /3, 7, 
and which is bounded by a line s. Then, by Stokes' theorem, 

f(p cos a + q cos j8 + r cos y)da = \ f(dx + Tjdy + dz). (23) 

The field around the charged sphere after the removal of the 
wire is perfectly symmetrical. Now, if we take for a a portion 
of a concentric sphere, the surface integral represents, apart 
from a constant factor, the quantity of electricity which passed 
through this surface and is thus equal to the charge of the portion 
of the spherical conductor which is cut out by the cone subtended 
by s and having its vertex at the centre of the sphere. The same 
charge must also be represented by the line integral in (23). 
We begin with taking for cr a small portion of the concentric 
sphere and we let this slowly increase. (The successive boundary 
lines s may, e.g., consist of a system of parallel circles.) Then 
the surface integral will continually increase, while this is 
impossible for the line integral after a has become greater than 
a hemisphere. Ultimately when the boundary s dwindles to a 



ii MECHANICAL AETHER THEORIES 39 

point, the surface integral represents the charge of the whole 
sphere, while the line integral becomes nil. The point is that, as 
is shown by (23), it is for purely kinematical reasons impossible 
to have at each place on the surface of the sphere a rotation 
around the radius which, seen from outside, has everywhere the 
same sense. The difficulty arises from the fact that the rotation 
is throughout solenoidal, while the dielectric displacement has 
not this property wherever there are charges. 

Larmor, Reiff, and others tried to save the theory by giving 
up the symmetry around the sphere. They assumed that at the 
place where the wire originally was the conditions are somewhat 
different from the remainder of the sphere, viz. that there is a 
canal K at that place within which the aether is loosened from 
the surrounding aether, so that while the aether in K is fixed, 
the surrounding one acquires rotations corresponding to the 
electric force, with the result that the line integral of the aether 
displacement along a path embracing the canal is equal to the 
whole charge of the sphere. It should be possible to keep up 
such a state by applying to the aether outside K, all along 
the surface of the canal, appropriate tangential external forces. 
Instead of this, one might think of attaching the twisted aether 
to that within K, which would prevent a complete untwisting 
of the medium outside of K. No objection can now be derived 
from (23), since in this equation f, ?;, f are assumed to be 
throughout continuous, while this condition does not hold at the 
surface of K. 



Ill 

KELVIN'S MODEL OF THE AETHER 

17. KELVIN'S MODEL OF THE QUASI-RIGID AETHER 

KELVIN conceived a model of a quasi-rigid aether built up of 
gyrostats. This is a complicated problem. It amounts to 
finding a system which permits all deformations but resists 
such as are associated with rotations and no others. The 
idea occurred to him to meet this requirement by means of 
gyrostats, for these oppose themselves to any change of the 
direction of their axes. Thus his task was to find a system con- 
taining a number of lines which remain parallel to their original 
direction at every deformation devoid of rotation, and which 
change their direction as soon as rotations are produced in the 
system. Along these lines one had then to lay bars bearing 
gyrostats. 

Let us consider a homogeneous deformation in which, that is, 
the components of the displacement of any of the points of the 
system, jf, 17, f, are linear functions of its co-ordinates x, y, z, 



77 - a 2 + a 2l x + a 22 y + a 23 z, 



If the coefficients a u , etc., are chosen arbitrarily, the deformation 
thus expressed is in general associated with a rotation. This 
can be found geometrically by recalling that a sphere is trans- 
formed into an ellipsoid and by determining those mutually 
perpendicular diameters of the sphere which correspond to the 
axes of the ellipsoid. The algebraic representation is found by 
rewriting the formulae thus : 

40 



Ill 



KELVIN'S MODEL OF THE AETHER 

% + a n x + |(a 12 - a 2l )y + (a 13 - a 3] 



41 



2 ( a ! 



- a 



32 



+ a 



a 32 - 



2 ( 




Thus the deformation is split into a shift, a dilatation, a 
rotation, and a shear, and we see that the conditions for an 
irrotational deformation are a ]2 =a 21 , 2 3 = 32 , O"$i = a>w In such 
a deformation, therefore, six coefficients, apart from the shifts, 
still remain undetermined. 

Now, to arrive at Kelvin's model, we construct in a plane 
a system of congruent equilateral triangles fitting to each 
other and erect upon these 
triangles as bases, omitting 
every second (as shown in 
the figure), regular tetrahedra. 
The corners of these tetra- 
hedra lie again in a plane and 
form a system of points such 
as those of the ground-plane. 
On these points, therefore, 
such a system of tetrahedra 
can again be constructed. 
This we do so that the bases 

of these tetrahedra should have the same position as those 
in the first layer of tetrahedra, so that the former can be 
obtained from the latter by a mere shift. That is to say, if 
the non-shaded triangles of the figure were first chosen as bases, 
we now take as bases the triangles PQR, RST, etc. In this 
way we can proceed, and similarly the system can be in- 
definitely continued on the other side of the ground-plane. 
Thus every corner point of the system will be the common vertex 
of four tetrahedra, at which also twelve edges will meet, two by 
two being prolongations of each other. Kelvin imagines now 
placed at every corner point a ball from which issue six bars and 
as many tubes, all of these being free to assume any direction 
whatever. The bars of one ball are now put into the tubes of 
other balls wherein they can be freely shifted back and forth. 




FIG. 12. 



42 AETHER THEORIES AND AETHER MODELS m 

This system can now be built up in the form of tetrahedra as 
just described. It will not oppose itself to any pure [irrotational] 
deformation. 

In the next place Kelvin introduces into each of our tetrahedra 
a system of three rigidly connected and mutually perpendicular 
bars of variable length (which may again be accomplished by the 
bar-and-tube method), and whose ends must remain in the grooves 
of the bars of the first system, so that they join each time two 
opposite edges of the tetrahedron. In a regular tetrahedron 
these bars coincide with the lines joining the mid-points of the 
opposite edges, but also in any tetrahedron whatever a set of 
mutually orthogonal intersecting lines, joining pairs of opposite 




A' A K C C 

FIG. 13. 

edges, can always be assigned.* The introduction of these 
systems of bars does not prevent any deformation. We will 
now prove that with an irrotational deformation of the original 
system the new bars are always shifted parallel to their initial 
direction. 

We first consider the case in which a regular tetrahedron 
undergoes an infinitesimal dilatation in the direction of one of 
the edges, say AC, which leaves the plane passing through BD 
and E, the mid-point of AC, in its place. It is enough to 
prove that a set of mutually perpendicular joins of opposite 

* To see this, notice that through a given point one and only one line can 
always be drawn which joins two skew lines. If, therefore, P bo an arbitrary 
point within a tetrahedron, there are through P three determined lines, each 
of which cuts a pair of opposite edges. By requiring these linos to bo mutually 
perpendicular we have three equations for the co-ordinates of P. Their 
solution for the case of a tetrahedron which differs but infinitesiraally from a 
regular one is implied in the following considerations of the text. 



m KELVIN'S MODEL OF THE AETHER 43 

edges of the new tetrahedron can be found which are parallel to 
the joins of the mid-points of the original tetrahedron. For this 
will then be the only existing set. Now, leaving the line EH 
unchanged, let us consider FO. If this is to remain parallel to 
its original direction, and still cut the line EH, its intersection 
point with the plane ADC must remain on the line E F and thus 
be shifted along it, say to F r . It is thus enough to show that 
the line through F' parallel to FG cuts BC' or, if 6?' be the point 
where that line cuts EG, that F F' = GG'. Now, this follows from 
the congruence of the triangles DFF' and EGG' . Similarly for 
the line joining the mid-points of AB and CD ; in view of 
the symmetry, it is obvious that this line will again be shifted 
along EH, so that we find, in fact, three orthogonal joins of the 
opposite edges, parallel to the original ones. What was j ust proved 
for a dilatation in the direction of the edge AC holds, of course, 
for the remaining edges, and our proposition will be completely 
established by showing that every irrotational deformation can 
be obtained by the superposition of six dilatations along the edges 
and of a displacement of the whole system. This, however, follows 
directly from what precedes, since an irrotational deformation is 
just determined by six independently prescribed ones. More- 
over, the relation between the values of the dilatations and the 
coefficients a n , etc., in the general formulae can readily be 
established. In fact, if a lf /3 V yi be the direction cosines of 
the first edge, d the dilatation in this direction, and similarly 
with changed suffixes for the remaining five edges, we have 
(apart from the shift) 

+ d z a 2 (xa 2 + yfa + 2y 2 ) + . . . 



whence 



44 AETHER THEORIES AND AETHER MODELS m 



wherewith the efs are determined, if the a's be given.* 

If now at the middlemost bars an arrangement is made of 
such a kind that for rotating the bars a couple is necessary 
which, for an infinitesimal rotation, is proportional to the rotation 
and has its axis coinciding with the axis of the rotation, then 
there is still no force opposing an irrotational deformation of the 
first set of points. But if there is a rotation, a resisting couple 
is produced which is proportional to the rotation and whose 
axis will always concide with that of the rotation, provided 
the three middlemost bars have all the explained property. 
To see this we have only to consider one of the tetra- 
hedra. Let OP, OQ, OR be the directions of the inner bars, 
and let us consider an infinitesimal rotation </> whose axis makes 
the angles X, /z, v with the bars. This can be resolved into a 
rotation <f> cos \ about OP, and <j> cos p, <f) cos v about OQ and 
OR. In the first rotation OP remains in its place, but for the 
rotation of OQ a couple is necessary of moment (7</> cos \ 
and with axis along OP. Similarly for the rotation of OR, 
making in all 2C<f> cos X. In quite the same way the rotation 
about OQ requires a couple 2C(f> cos /*, directed along OQ, and 
that about OR a couple 2C< cos v. Compounding all these 
couples we have, as announced, the couple 2(7< whose axis 
coincides with that of the rotation. 



18. SOLID GYROSTAT 

As a first device to make a couple necessary for changing the 
direction of a bar Kelvin proposed the ordinary gyrostat. The 
bar AB (Fig. 14) carries a fixed ring in which a second ring is 
mounted, free to rotate around PQ JL AB. The diameter RS of 

* In fact, it can be shown that the determinant of the coefficients of 
d l9 . . dt does not vanish, so that the equations are compatible with each 
other. In proving this the co-ordinate axes can bo chosen arbitrarily. If for 
these the joins of the mid-points of the opposite edges are taken, one of the 
direction cosines for each edge is zero, while the others become l/\/2 with the 
same or opposite signs. The absolute value of the determinant is then found 
to be equal J. 




ni KELVIN'S MODEL OF THE AETHER 45 

the second ring carries at the centre a fly-wheel with RS as axis. 
It is clear, first of all, that this arrangement does not oppose 
itself to a rotation of the bar AB around the axis PQ. Kelvin 
mounts then on every bar two such rings in mutually perpen- 
dicular planes. But let us first 
consider a single gyrostat. Its 
inner ring can spin about PQ, 
and the fly-wheel about RS. 
For every contemplated rotation 
and similarly for every moment 
with respect to an axis a certain 
sense will be assumed as the 
positive one, viz. that from which 

the rotation appears to be anticlockwise. Let for the rota- 
tions in question the positive sense be given by OP and OR. 
Let ft> be the angular velocity of the fly-wheel with respect to 
the inner ring and let AOR = 0, and, therefore, the angular 
velocity about OP. Further, let Q be the moment of inertia 
of the fly-wheel corresponding to the axis RS, and Q' that 
corresponding to the axis PQ. Suppose now that, while APE is 
kept fixed, the inner ring and the fly-wheel are spinning. Then 
Q'0 will be the moment of momentum of the fly-wheel along 
[about] OP, and its time-rate of change Q'Q. This change of 
the moment of momentum can be resolved (Fig. 15) along OR 
and OT perpendicular to OR and OP, and therefore in the 

plane AOR. Instead of the angular 
velocity o> and the moment of 
momentum Qu* about the axis OR, 
the fly-wheel has after a time dt 
the angular velocity w+da) and the 
moment of momentum Q(&>+da)) 
about the axis OR' (ROR f =d0). 
Thus the components of the moment of momentum are, up to 
terms of the second order, Q(a>+da>) along OR and Q&d9 along 
OT, and its rate of change Qu> about OR and Q&0 about 
OT. The couples produced by this change of the moment of 
momentum are due to the forces exerted by the inner ring on 
R and 8 which, however, can give no couple about OR. 
Thus, Qci = 0, whence 

01= constant. 




46 AETHER THEORIES AND AETHER MODELS m 

Let us now consider the system consisting of the inner ring and 
the fly-wheel. This is acted upon by forces applied at P and Q, 
which thus are unable to produce a couple about OP. If now 
q be the moment of inertia of the inner ring with respect to the 
axis OP, the moment of momentum of the whole system relatively 
to this axis is Q'8 + qQ, and since this cannot vary, itself is 
constant. 

Let us now see what happens when the bar carrying the gyro- 
stat is turned from its position A B Q (Fig. 16) to AB contained in 
the plane A^JB^ and making with A B Q an infinitesimal angle f . 
We will suppose that the bar is being kept in its new position 
and that the exterior ring is held in the original plane A^P^B^ 
so that OP lies in this plane perpendicularly to AB. The forces 
which through this change of position are brought to act upon the 

inner ring have their points of 
application at P and Q and can 
thus produce no couple about 
PQ. Let there be initially no 
rotation about OP and let 
RS fall initially into the direc- 
tion of ,4J3(0 = 0). Then we 




FIG. 16. shall have only the moment 

of momentum about OA Q 

amounting to Qa>, which we resolve into Qco along OA and Qcoe 
along OP. Now, suppose that owing to the said change of position 
the inner ring which originally did not spin acquires an angular 
velocity 0. (This will turn out presently to have a non- vanishing 
value.) This angular velocity is shared also by the fly-wheel, but 
the moment of its momentum about the axis OR does not 
depend on the angular velocity 6 but only on the angular velocity 
around OR, and the latter must thus still retain its original value 
&), because during the considered change of position the fly-wheel 
was acted upon only by such forces, at R and S, whose moment 
with respect to OR is nil. Thus, after the displacement, we have 
the moments of momenta Qco along OA and (Q f +q)0 along OP, 
and since there is no couple about OP, 

Qa>c = (Q'+q)6, 
whence 



in KELVIN'S MODEL OF THE AETHEE 47 

Through the change of position of the bar the innermost ring is 
thus set spinning, the direction and the velocity of this spin 
being determined by the last formula. The subsequent course 
of things can be found by applying the considerations of the 
early part of this article (since these hold for every position of 
the bar AB). We can thus conclude that the acquired angular 
velocity will remain without change of size, so that the angle 
will gradually mount to considerable values, and that the 
moment of momentum of the fly-wheel will undergo, per unit 
time, a change of which the component along OT, perpendicular 
to the plane of the inner ring, is QcoO. An equal couple is 
necessary to keep APB in the new position, and substituting 
for the value just found, this couple turns out to be 



Q'+q* 



and is directed along OT. Resolving it into components along 
OA and ON OA (corresponding to 6 = 90), we have, along OA 9 



and along ON, 

Q'+7 cos ' 

It is the latter component which is required for our theory. 
In fact, this couple whose direction coincides with that of the 
rotation from A B to AB is necessary to keep the bar in the 
new position, in other words, the bar resists this rotation 
with an equal couple of the opposite sense. The other com- 
ponent, proportional to sin 0, is due to the circumstance that 
some forces are also necessary to keep the plane ABPQ in its 
original position. To provide for this we can mount upon our axis 
yet another gyrostat in the same plane with and entirely similar 
to the first, with the only difference that its fly-wheel spins 
originally in the opposite sense. Then will have for the two gyro- 
stats always opposite values, and since the initial value of is 
for both, their angles themselves will be equal and of opposite 
signs. The couple about ON, necessary to reset the bar, will thus, 
of course, be twice as great. Similarly, the gyrostat which we 
have already introduced in the plane perpendicular to that of the 
first can be replaced by a set of two oppositely spinning gyrostats, 



48 AETHER THEORIES AND AETHER MODELS in 

so that ultimately each of the bars carries four gyrostats. Then 
the system resists every motion associated with a change of the 
direction of the bars and no other, exactly as was required. 
The serious objection against this arrangement (as an aether 

model) is that the couple required 
to keep the bar in the new posi- 
tion becomes, owing to the factor 
cos 8, smaller and smaller (and 
A" finally even negative). 




19. LIQUID GYROSTAT 

Yet another device was pro- 
FIG. 17. posed by Kelvin to ensure that 

a couple should be necessary for 

changing the direction of the bars, viz. a liquid gyrostat. This 
consists of a ring-shaped tube (Fig. 17) filled with a circulating 
liquid and free to rotate about one of its diameters as axis. 
This axis lies in one of our previous bars. Such 
a single ring has the same effect as the previous 
solid gyrostat. 

The theory is much the same as before. Let 
the axis about which the ring rotates be chosen 
as X-axis and let the position of the ring be 
determined by the angle contained between the 
normal of its plane, taken in 
the sense appropriate to the 
motion of the liquid, and the 
axis OZ. For = the ring 
would then have the position 
shown in Fig. 17, where the 
Z-axis is assumed to point 

forward. The actual position of the ring at any instant follows 
therefrom by a rotation about OX. Compare Fig. 18, where 
OZ' is the normal to the plane of the ring, while OY' is 
contained in that plane. 

Let the angular velocity of the liquid be o> and the moment 
of inertia of the liquid with respect to the axis of the ring Q, 
and that with respect to a diameter Q' (so that, the tube being 
very slender, Q' 





FIQ. 18. 



in KELVIN'S MODEL OF THE AETHER 49 

Consider first the liquid itself. This has at first the moment 
of momentum Q'6 about OX and Qco about OZ', and then 
Q'(8 + dO) about OX and Q(a)+do>) about OZ". The vectorial 
difference between the moment Q(co+da)) about OZ" and Qo> 
about OZ! can be resolved along OY' and OZ', giving as com- 
ponents of the rate of change of the moment of momentum Qti 
along OZ' and -Q&>0 along OY'. The liquid is subjected to no 
other forces than the pressure of the tube walls, and in virtue of 
a known property of surfaces of revolution that pressure gives 
rise to a system of forces all of which cut the axis OZ' and thus 
can give no couple about OZ'. Consequently, Q<w=0, and 
therefore CD = constant. To consider now the system made up 
of the liquid and the tube, let the moment of inertia of the 
latter relatively to the axis about which it can spin be q. 
Then the rate of change of tho moment of momentum of 
the system along OX will be (Q f + q)6, and since the system 
is acted upon only by forces exerted by the axle, this is 
again zero, so that is constant. Thus, while the axis OX 
is kept fixed, the ring can spin about it uniformly and 
at the same time the liquid can circulate in the tube with 
constant velocity. But to keep the axis in position we must 
apply to it a couple -QwQ about the line OY', for, as we saw, 
the latter is the rate of change of the moment of momentum of 
the liquid. 

Suppose now that initially = and $ = 0, so that we have 
to do with the case represented in Fig. 17. No work, of course, 
is required to turn around the bar in the plane XOY, but a 
rotation about OY is opposed by a couple. To see this, we 
consider a rotation about OY through an infinitesimal angle e, 
which brings the bar OX (Fig. 19) into the position OX' in which 
it is again kept fixed. The normal ON of the plane of the ring has 
now moved into the plane YOZ' and will first coincide with OZ' 
or at the utmost include with this direction an angle of the order 
of 6 ; it has acquired, however, an angular velocity about 
OX' which will differ from zero though it can only be small 
of the order of e (since it is the effect of a change of position 
determined by e). In fact, after the readjustment we have the 
moment of momentum (Q' + q)Q about OX' and Qa>' about ON, 
and resolving both along OX and OZ and keeping in mind that 
initially and vanished, we find for the increment along OX 

VOL. i E 



50 AETHER THEORIES AND AETHER MODELS m 

the value (Q f + q)6 + Qwe, and as for obvious reasons this must 
vanish, we have 

s\ 

0=-, 



Let us still notice that resolving the moment of momentum 
of the liquid after the rotation e along the axes OX, OY, OZ, 
the value of the last of these is found to be Qw', since terms of 
the order e 2 can be neglected. And since the forces exerted on 

the liquid during the change of the 
direction of the bar give no moment 
with respect to OZ, ' must be 
equal to A?. 

The value found for is the 
angular velocity acquired by 
the ring through the change 
f portion of the bar, with 
which it continues to spin. 
For the subsequent motion holds all that was just said about 
the possible motions in the original position of the bar. If OY 
and OZ in Fig. 18 are taken to represent the directions denoted 
in Fig. 19 by OF and OZ', it will become manifest that the bar 
must be acted upon by a couple about OY' amounting to 




which can again be resolved into e cos along OY and 

Q 2 o> 2 ^ +? 

% esinfl along OZ. The rotation of the bar about OY 

Q +? 

requires thus first of all a couple about OY proportional to the 

rotation. But in addition to this another couple, about OZ, is 

required which can again be avoided by mounting upon the axle 

two gyrostats with liquids circulating in opposite senses. Then, 

of course, the required couple will again be doubled, i.e. amount to 



The difficulty pointed out in connection with the solid gyrostats 
exists also in the present case. These models can therefore be 



m KELVIN'S MODEL OF THE AETHER 51 

used only if the bar carrying the gyrostat is not being displaced 
too long in the same direction, as e.g. in the case of periodical 
motions of small amplitude. 

We will consider the case in which e is a periodic function 
of the time (and remains very small). Then the same property 
must also hold for the couple determining the position of the 
bar. This vibration about the axis OY is then associated with 
an oscillation of the ring-shaped tube around the bar. In other 
words, also will be a periodic function of the time and the 
deviations will remain small. Moreover, since the position 
changes of the bar are very small, these oscillations of the tube 
can be considered as taking place about the axis OX (Fig. 19). 
Thus we have the following moments of momenta : (Q 1 + q)0 
about OX, (Q'+q)t about OY, and Qco about the axis ON 
normal to the plane of the ring. This axis changes continually 
its position and makes at a given instant the angles e, - 6, 
with the axes OX, OY, OZ. In considering the moment Qw 
about ON, the latter axis cannot be replaced by OZ, for though 
these directions differ but very little, yet the moment of momentum 
itself is not small. This, therefore, must still be resolved along 
OX, OY, and OZ. The component along OX is Qve, that along 
OY, -Qco0, and that along OZ, Qw. Let us now suppose there 
were no other external forces than the couple producing the 
oscillations of the bar in the plane XOZ, which therefore has OY 
for its axis, and let the moment of this couple be K which, of 
course, is a periodic function of the time. We shall then have 
the following equations of motion : first, by considering the 
moment of momentum about OX, 



the zero on the right hand being conditioned by the vanishing 
initial values of e and 0, and, second, by considering the moment 
of momentum about OY, 



while the third equation of motion expresses simply that o> is 
constant. Eliminating 0, 



52 AETHER THEORIES AND AETHER MODELS m 

This has the form of the equation of forced vibrations. We can 
also speak of proper or free vib.*ations of our system, which may 
exist when K -0. The frequency of these free vibrations is 



Q' + q 
and their period 

2<rrQ'+q 
w Q" 

In presence of an external force whose period is large 
compared with that of the free vibrations, the state at any 
instant will coincide with that in which the system would be in 
equilibrium under the action of the force prevailing at that 
instant. If the period of the external force is small, the phase 
difference between the vibrations of the system and of the force 
is -|TT. If the liquid is assumed to circulate in the tubes very 
rapidly, we may always limit ourselves to the first case. Mount- 
ing then on each of the inner bars four gyrostats, in a manner 
already explained, we can represent the aether quite well. 
This can also be so arranged (by varying Q or CD) as to yield a 
non-homogeneous medium, by means of which also the refraction 
and reflection phenomena can be represented. For the free 
aether the period of the external couple K must then always be 
assumed very long as compared with that of the free vibrations, 
and thus also with the time of revolution of the liquid in the 
tubes. Otherwise the term with e in our equation would come 
into prominence, and the aether would have to be given different 
densities for vibrations of different frequencies, whicli in turn 
would influence the velocity of propagation. 



20. LIQUID IN TURBULENT MOTION AS AETHER MODEL 

Kelvin tried also to represent the aether by means of an 
incompressible liquid in turbulent motion. The dimensions 
characterising this medium are those of the vortices. Upon 
these a coarser motion can be superposed, as e.g. a pro- 
pagation along the 7-axis of transversal vibrations in the 
XOY- plane. If /(y,0 be the velocity of a particle due to 
this vibration [along the JST-axis] and u' t v', w' the velocity 



m KELVIN'S MODEL OF THE AETHER 53 

components of the turbulent motion at the point in question, 
the resultant velocity is u =f(y , t) + u' , v=v' 9 w=w'. The 
velocity corresponding to the vibration can be obtained separ- 
ately by averaging over spaces whose dimensions are small 
compared with the wave-length of the oscillatory motion 
but large compared with the vortices. The average velocity 
is then nil for the turbulent motion, while f(y,t) is nearly equal 
throughout the domain over which we have averaged. The 
averages thus defined will be expressed by bars over the letters 
representing the magnitudes in question. We will now deduce 
two equations which will show that transversal vibrations can, 
in fact, propagate themselves and will give us also their 
propagation velocity. The first of these is arrived at by con- 
sidering the momentum along the X-axis carried across a 
plane perpendicular to the Y-axis. In fact, such a plane is 
traversed by the liquid in either direction and even in equal 
amounts, but the liquid flowing in one direction can have 
a different momentum from that streaming in the opposite 
direction. For the excess of momentum transferred towards the 
positive over that in the negative direction we find, per unit 
area, p u'v\ if p be the density of the liquid.* By considering 
the increase of momentum within a short cylinder having its 
faces perpendicular to the Y-axis, we find as the first of the 
required equations 



Next, we have the usual equations of motion of a non-viscous 
liquid, 

du du du du dp\ 

--- -- 



/nf .. 
(25) 
v ; 



where p is the pressure divided by the density. 

* This can bo compared with the momentum transfer in a gas due to 
the passage of molecules through a plane, as considered in the theory of 
viscosity. 




54 AETHER THEORIES AND AETHER MODELS n 



Substituting our special values, w=w'+/(y,), etc., the firsl 
two equations become 



u-~-+v~- 
ox oy 

(V ,. 4 JDv' ( ,W frf JM 

- - - l tt S +v a y +* 



u' ,du' dp] 

-+w' ~ +J-\ 
y oz dx) 



whence 

o 

dt 



,3(uV) ,3(ii 

'- v ^ '+D' -+u ^ +v f u -' 

ex cy vz vx cy) 



Averaging both sides, the equation can be considerably simplified, 
Both /(if, t) and 3/(y,0/^ can be considered as constant over the 
averaging space. Thus, 



(it 
since y' = 0. Again we have -$ = for every magnitude < which 

f'33 

fluctuates over distances small compared with the dimensions 
of the averaging space. We have still to take care of terms 
such as u'd(u'v')ldx. If this be written 

,d(uV) 3ii'(ttV) W 

u-~ ' = \ 7 -f*V n , 
dx Sx ux 

it will be seen that in the process of averaging we are left with 
the average of the expression 



-s- n . , 
dx ty dz/' 

but this is everywhere zero [the liquid being incompressible], 
Thus our equation becomes 



in 



KELVIN'S MODEL OF THE AETHER 



55 



Now, let pop be the pressur| when f(y,t) is throughout zero, 
and pQp+p'p the actual pressure, and let us assume that 
f(y,t) is always very small in comparison with the velocities 
of the turbulent motion. In absence of vibrations the left-hand 
member of (26) and the first term of the right-hand member are 
zero, so that 



vx 



vy 



We will assume that this equation holds also when u' and / are 
taken to stand for the velocities of the turbulent motion accom- 
panying the vibratory motion. Then we can put 



dy 



(27) 
x ' 



A weak point of this theory, however, is, among others, that one 
does not know whether the properties of the turbulent motion 
remain unaltered by the vibrations. But this and similar 
difficulties need not detain us, our only purpose being to describe 
Kelvin's scheme in its main lines. 

In order to transform (27) we return to the equations 

du 
dt~ 

dv 

di" 

dw 

ar 

from which follows : 

dw du du dv\ 

If now u, v t w are replaced by the values they have in the presence 
of vibrations, f(y,t) affects only the term with dujdy, and thus 
we find 



?M 


to 


to 


& 


^ + 


% + 


W W Z ~ 


?JC 


to 

1 


to 


to 


Vp 


V 


% + 


w '(h 


% 


tko 

j 1 


dw 


flW 


dp 


f ac 


<v 


W 7>z = 


~fo> 



(25) 



The solution of this equation can be written symbolically 



56 AETHER THEORIES AND AETHER MODELS m 

The meaning of the symbol A" 1 is known from the theory of 
Poisson's equation A0 =^, whose solution is 



In our case, owing to the rapid changes of sign of dv'fix, while 
df(y,t)ldy varies much slower, only the nearest neighbourhood of 
the point for which p' is to be evaluated contributes to the space 
integral. Consequently, df(y,t)ldy can be regarded as a constant 
factor, so that 



-. 

r dy dx 

Putting 

Jf -+&-<*, 

dx cy ^ 
and keeping in mind that df(y 9 t)jdt varies slowly, we find 



-V. . . (28) 

. v ' 

Now, by averaging, 



and, though this is not quite correct, we may assume that also 
the third mean, 



has the same value. Thus, 



A- V =i*/AA- V = Jt/ 2 . 



To find the last term of (28), we put -A~ V = ^. Then 



dx dx 



m KELVIN'S MODEL OF THE AETHER 57 

In averaging, the first term contributes nothing, so that 

' ~~dw'\ t ,dv' 



and by reductions similar to those just made, 



If /2 2 be the mean square of the velocity in the turbulent motion, 
w' a + v' 2 + w'* - Si/ 2 = 7^2, and (26) becomes 

. . . . (29) 



^ 
dt 

This is the second of the required equations. Combining it with 
our first-found equation, 

,1(W)_ <>f(y,t) 
~ty dt'' 

let us eliminate u'v'. This gives 



which shows that in our liquid transversal vibrations can be 
propagated, with the velocity 



IV 

ATTRACTION AND REPULSION OF PULSATING 
SPHERES 

21. NATURE OP THE PROBLEM 

CONSIDER an incompressible frictionless liquid in which spheres 
are moving about. This motion may be either a translation or 
a swelling and shrinking of the spheres, both of which may be 
periodic. 

We will first take up the problem of finding the motion of 
the liquid if that of the spheres be given. 

Particles of a non-viscous incompressible liquid which once 
do not rotate will never rotate. There exists then a velocity 
potential, and the velocity components can be represented by 

3* fXD 3O 

U = -~- V = -X W = -7T- 

ox vy dz 

Owing to the incompressibility 4> will satisfy Laplace's equation 
A<1>=0. Whence it follows that the velocity at every point of 
a space-region will be determined, if at every point of its boundary 
the normal velocity is known. For then 94>/Pw all over the 
boundary is known, and thus also <1> throughout that space- 
region is determined, since it is continuous and satisfies Laplace's 
equation. Let us consider the space limited by a fixed closed 
surface at infinite distance and by the surfaces of all the spheres. 
Then the normal velocity all over these boundaries is known, 
the motion of the spheres being given. If a, 6, c be the co- 
ordinates of the centre of one of the spheres and R its radius, 
the motion is determined by a, 6, c, and ft. It remains to find 
a suitable solution of A4> =0. This will be supplied by 



68 



iv PULSATING SPHERES 59 

where h l9 A 2 , , . . h k are measured along arbitrarily chosen 
directions [and r is the distance from the centre of the sphere]. 
This is a homogeneous function of the space co-ordinates of 
degree - (k + 1). If we put 



then (the spherical harmonic function) Y k is a homogeneous 
function of the zeroth degree. Next, write 



then H k will be a homogeneous function of the k th degree 
which can be represented by a whole algebraic function of the 
co-ordinates. For at every differentiation the highest exponent 
of r in the denominator is increased by 2, so that the highest 
exponent in V k is 2 + 1 , and the remaining ones are by an even 
number lower. Thus in H k all denominators fall out, while in 
the numerators only even powers of r occur. H k also satisfies 
the equation of Laplace, but it cannot be used for representing 
the velocity potential in a liquid extending to infinity, since it 
becomes infinite at an infinite distance. On the other hand, V k 
cannot be used if the origin of r lies' within the liquid. ' 

22. A SINGLE MOVING SPHERE 

Consider the space outside a single sphere. Then the simplest 
solutions are, for A = 0, F = l/r, y o = l, H Q = l. If we take 
cl> = C/r, the velocity will be radial and equal to - C//" 2 . By an 
appropriate choice of C as a function of the time this solution 
can be adapted to a sphere with fixed centre and variable radius R. 
In fact, since at the surface jft = - 0//2 2 , we have 



= - 

r 2 

For k = 1 , the next simple case, we find, with the X-axis along 
the direction of h, 

T7 J) /1\ x-a v x-a u 
F i--fcU- r" Y i = r .*i-*-* 

If we take for the velocity potential outside the sphere 
n_C(x-a)_C(x-a)l 

Nr n "* ~~ " ~ t>> 

r 3 r r 2 



60 AETHER THEORIES AND AETHER MODELS iv 
then, at its surface, 

3O o^-a ! 2CcosA 
cto" 20 R & IP ' 

This can be adapted to the case in which the sphere has a 
translation velocity along the -X-axis. For then the component 
of this velocity along the surface normal is also proportional to 
cos \. Thus, 

-2(7 cos A . , 
3 = a cos A, 



whence 

and O=-|d# 3 ..... (31) 

Since (x-a)// 8 arises from the differentiation of 1/r, this motion 
of the liquid can also be obtained by a superposition of the 
motions due to two spheres devoid of translation but pulsating 
in opposite phases and placed at an infinitesimal distance from 
each other. 

Let now the liquid be in a certain state of motion and let 
us ask how this motion is disturbed by the presence of a sphere. 
It will appear that due to the latter a certain state of motion is 
superposed over that already existing. We will call this the 
reflected motion. It will be the weaker, the farther away from 
the sphere. 

23. A SPHERE AT REST IN A LIQUID WITH A GIVEN 
MOTION 

Let us consider the case in which the disturbing sphere is at 
rest. We choose its centre as the origin of co-ordinates. The 
[irrotational] motion of the liquid being given, so also is the 
velocity potential 4> which can be developed around the origin 
into 



where the values of the derivatives of 4> prevailing at the centre 
have to be taken. The state of motion thus expressed can be 
interpreted as if it were due to a superposition of different 



iv PULSATING SPHERES 61 

motions whose velocity potentials are represented by the succes- 
sive groups of terms. Thus, a velocity potential <l> represents 
rest, the next group, 

ao ao ao\ 



expresses a constant flow with the velocity prevailing at the 
centre, and so on. This series development can also be put 
into the form 



The reflected motion, combined with the original one, must give 
all over the surface of the sphere the radial velocity 0. Let us 
put for the reflected motion 



and let us suppose that to each of the motions to be compounded 
there corresponds a reflected motion. Then we must have at 
the surface, for every k separately, 



Consequently, 



and the velocity potential of the reflected motion becomes 



24. n SPHERES 

Let us now turn to the problem of finding the state of motion 
due to the arbitrarily prescribed motion of n spheres, of which 
the positions, that is, as well as the sizes and velocities, are given 
for all times. The velocity potential <f> must then satisfy 
Laplace's equation and the condition that at each of the surfaces 
dtf>/dn should have given values. We shall find 4> by super- 
posing n solutions, each corresponding to the motion of a single 
sphere. Each of these solutions will be found by proceeding 
first as if the sphere in question existed alone and then by taking 



62 AETHER THEORIES AND AETHER MODELS iv 

account, in an obvious way, of the repeated reflections of the 
state of motion produced by thai sphere. Suppose, for instance, 
there were only two spheres, the first of which has a given 
motion, while the second is at rest. Let, in terms of the 
velocity potential, 4> x be the motion due to the presence of 
the first sphere alone, <f> n the motion arising through the 
reflection of 4> x at the second sphere, 4> m the motion produced 
by the reflection of <J> n at the first sphere supposed to be at rest 
in its instantaneous position, and so on. Then the actual motion 
will be found to be represented by the infinite series 



which will converge the more rapidly the smaller the spheres in 
comparison with their mutual distance. This series gives the 
solution of the problem. For the equation of Laplace is satisfied, 
since each of the terms does satisfy it and the derivatives of 
the series can be found by differentiating separately its terms, 
and all boundary conditions are satisfied. In fact, at the surface 
of either sphere r)<I>/?n is expressed by a series whose terms, as 
far as the second sphere is concerned, cancel each other in pairs, 
while for the first sphere the first term only survives and this 
has the value which is prescribed for its surface. 

25. Two SPHERES 

We will now determine the forces which two pulsating spheres 
exert upon each other. The equations of motion 

du (hi du\ 



become, in terms of the velocity potential, 

dp a rso 



A 
> etc -> 

and since p is constant, these give, with V written for the velocity, 

. . (32) 



where C is a constant. This, however, can be omitted, since a 
pressure which is constant all over a surface cannot produce 
any motion. 



iv PULSATING SPHERES 63 

Let 12 1 and R 2 be the radii of the spheres and I the distance 
of their centres. Then the velocity potential due to the motion 
of the first sphere is, by Art. 22, 



where r' is the distance from the centre of this sphere. 

If now the origin of co-ordinates be placed into the centre of 
the second sphere and if the Jt-axis be laid along the produced 
join of the centres, the last expression can be developed around 
the second sphere into the series 



. . .], 



of which it will be enough to retain the first two terms, as 
according to our assumption the radii of the spheres are small 
compared with L Since the first term is constant, the motion is 
determined by the second term. At the second sphere a reflected 
state of motion is produced, for which the velocity potential is, 
by Art. 23, 



where r denotes the distance from the centre of the second sphere. 
Rigorously we should take account also of the motion arising 
from this through a reflection at the first sphere, and so on. 
But since these reflected motions become rapidly weaker, they 
may be disregarded. The velocity potential due to the motion 
of the second sphere is -7Z a 2 $ 2 /r which, to the desired degree 
of accuracy, need not be supplemented by any reflected motion 
at all. For our purpose is to determine the force experienced 
by the second sphere. This, however, requires only the know- 
ledge of the motion of the liquid in the neighbourhood of the 
second sphere, and the reflection of the last-mentioned motion 
from the first sphere would contribute only an expression con- 
taining a factor of the order IIP. Let us still put 



then the whole velocity potential in the neighbourhood of the 
second sphere will be 

Ida* 



64 AETHER THEORIES AND AETHER MODELS iv 

By (32), to determine the pressure, d/dt is required. Now, 
since in differentiating with respect to the time x, etc., are kept 
constant, 

SO _ 1 / x\ <P ai ^x ( cPa, da l da*\ I d*a 2 
dt " " I V " l) W + WV* 1 dP + dt 'dt / r dP ' 



To find the force upon the second sphere, we have to integrate 
over its whole surface, but in doing so we may disregard the 
term F 2 in (32).* Again, all terms not containing x can be 
omitted, since these represent a pressure uniform all over the 
surface which, therefore, as was just mentioned, gives rise to no 
force upon the sphere as a whole. We are thus left with 

(Pa, 3 / d*a, da, da 2 \ 



{3 riPaj 1 da l da 2 ] 
~H2/ 2 dP + 2fti 1 dt dt}' 



Now, a pressure p == - cxp gives for the ^-component of the force 
upon an element da- 

+ cpR 2 cos 2 ft/or, 

whence, the force upon the whole sphere, 

JcpR 2 cos 2 Oder = $7TCpR z * = 4:7TCpa 29 
and substituting the value of c, 



We will suppose that the spheres pulsate rapidly, so that the 
perceptible force is the average of this expression over a time 
interval comprising many periods. Moreover, let both spheres 
have the same pulsation period. Total derivatives with respect 
to the time can then be omitted, so that the last expression 
for the force upon the second sphere can be written 



* For, evidently, we can limit ourselves to that part of JF 2 which 
depends on the motion of the first as well as on that of the second sphere, 
that is to say, to the scalar product of the velocity-vector F lf which would 
correspond to the motion of the second sphere alone, into r,, due to that of 
the first sphere only. Now, at the surface of the second sphere F s is radial 
and F! tangential, so that their scalar product vanishes. 



iv PULSATING SPHERES 65 

and the required average ultimately becomes 



~dt dt ' 

Whence we see that, if the pulsations of the two spheres are 
in phase, this force is negative, that is to say, we have an 
attraction, and if their phases are opposite, a repulsion. Let us 
consider this in more detail for the case of simple harmonic 
oscillations. If the radius varies periodically, so also does the 
volume of the sphere, and with the same period, and if the 
changes remain small, a simple harmonic variation of the radius 
is associated with a simple harmonic variation of the volume. 
We can write, therefore, 

0,1 = A! + G! cos (nt + e^, 
where c v is small compared with A l9 and similarly, 

<i 2 == A 2 +c 2 cos (nt + 2 ). 
This gives for the averaged force along the -3f-axis 



- _______ - ______ 

K x = - -w-c L c 2 n 2 sin (nt + x ) sin (nt + 2 ). 

In order to find the mean over a long time, it is enough to 
average over a full period. Now, since 

2 sin (nt + e^ sin (nt + 2 ) = - cos (2nt + l + 2 ) + cos (ej - e 2 ), 
the required average turns out to be 

cos ta - e 2 ). 



Thus we have an attraction for e l - e 2 < 7r/2, reaching a maximum 
for j = 2 , and a repulsion for e l - 2 > Tr/2, attaining a maximum 
for ei - e 2 7r - 



26. TREATMENT OF THE PROBLEM BY MEANS OF 
LAGRANGE'S EQUATIONS 

The results just found might also have been deduced from 
Lagrange's equations of motion. In using these we will introduce 

VOL. I F 



66 AETHER THEORIES AND AETHER MODELS iv 

as co-ordinates, also for the case of a system of any number of 
spheres, the Cartesian co-ordinates a, 6, c of the centres and the 
radii R of the spheres. The kinetic energy T is then a quadratic 
function of o, 6, c, tt with coefficients which are functions of the 
co-ordinates. If X be the component of the external force upon 
one of the spheres, taken along the X-axis, we have 



_ 
fi\aa/ <V 

The kinetic energy T can be split into two parts, one due to 
the liquid, the other due to the spheres, and in accordance with 
this the inner force keeping equilibrium to the force X can be 
divided into two parts. We then find for the force exerted upon 
the sphere by the liquid along the X-axis 



<f /MV\ 

dt\<>dJ 9 



if T v be the kinetic energy of the liquid. The latter is (owing 
to 



the last integral to be extended over the surface of all the spheres 
(it being assumed that we need not reckon with the infinitely 
distant boundary), and the normal to be taken towards the liquid, 
i.e. away from the spheres (hence the negative sign). The last 
integral can, of course, be divided into parts, each extended 
over the surface of one of the spheres, thus : 



In the case of two pulsating spheres we have for the second 
sphere 9<J>/9w= < B 2 and therefore, 



ft 1 * 

= l{ -I'd 



saa -R a 



where a v a 2 are as explained in the preceding article. On in- 
tegrating, the second and the third terms contribute nothing, 



iv PULSATING SPHERES 67 

while the remaining ones are constant, and we are thus left 
with 



Similarly the integral over the first sphere is 



The symbol a appearing in (34) stands in this case for our Z, and 
the first term of that expression gives 



This is exactly the force we have already found. With regard 
to the second term in (34), notice that what we require are the 
forces which act on the spheres when these have only a pulsating 
motion, while their centres are fixed. Thus I ! =0. In evaluating 
the first term of (34) we can, for obvious reasons, simplify T y by 
putting /=0, as we did in fact. In the second term, however, 
this simplification can be introduced only after the differentiation 
with respect to /. The evaluation of this term, which would 
lead us to the first term of (33), would thus require a further 
consideration of T n into which, however, we need scarcely 
enter, since we are concerned only with the time derivative of 
dT v pd, and the average of 4 such a derivative over a full period 
is nil. 

27. PEARSON'S THEORY 

Consider the case of an infinitely long period [of pulsation], 
that is to say, a sphere which goes on expanding for ever. Then a 



surface enclosing the sphere is traversed by the amount f -^.-1. ) 

or ^Trdajdt of fluid. Let this be denoted by e v Then the 
attraction between two such spheres will be /oe 1 e 2 /47rP. 

Now, Pearson abolishes the spheres and supposes that there is 
towards certain points an incessant stream of fluid (aether sources 
or sinks). It is true that one cannot well picture to himself such 
a state of things. But as we can imagine points spread over 



68 AETHER THEORIES AND AETHER MODELS iv 

a surface, towards which the aether streams from the sur- 
rounding space, to spread itself then over the surface, we 
should also be able to place such aether sources in three- 
dimensional space by calling to our aid the four-dimensional 
space. Similarly, there would exist points (aether sinks) where 
the fluid is being annihilated.* Two such aether sources would 
then attract each other, and similarly two aether sinks, while a 
source and a sink would repel one another. 

Apart from the strangeness of such a representation there is 
also another objection. In fact, if the expanding sphere be 
entirely omitted, there would still be a force upon the place 
whence the aether emanates. If a small sphere be described 
around the aether source, the force exerted by the aether will 
be a pressure upon the sphere, and one would have to imagine 
that the source is being displaced together with the sphere. 

There is a striking difference between the present case and 
that of electrical actions. For here we have attraction between 
points [sources] of equal, and repulsion between points of opposite 
signs. 

A similar theory was proposed by Korn to account for 
molecular forces and gravitation. He imagines a number of 
pulsating spheres, all in phase with each other ; the latter 
coincidence is secured by enclosing the whole space in a limiting 
surface which is acted upon by a periodical external force. This 
is propagated instantaneously through the incompressible fluid 
and makes the volumes of all the spheres alternately increase 
and diminish in the same phase. 



In what precedes a description was given of some of the 
attempts which were made in order to account for various 
phenomena, and especially the electromagnetic ones, by means 
of speculations about the structure and the properties of the 
aether. To a certain extent these theories were successful, but 
it must be admitted that they give but little satisfaction. For 
they become more and more artificial the more cases are required 
to be explained in detail. Of late the mechanical explanations 
of what is going on in the aether were, in fact, driven more and 
more to the background. For many physicists the essential 

* [For the three-dimensional beings, that is.] 



iv PULSATING SPHERES 69 

part of a theory consists in an exact, quantitative description of 
phenomena, such e.g. as is given us by Maxwell's equations. 

But even if one adheres to this point of view, the mechanical 
analogies retain some of their value. They can aid us in thinking 
about the phenomena, and may suggest some ideas for new 
investigations. 



SOURCES AND REFERENCES 

G. G. Stokes, On the Aberration of Light, Phil. Mag. (3), vol. xxvii., 1845, p. 9. 
H. A. Lorentz, De aberratietheorie van Stokes, Zittingsverslagen Kon. Akad. v. 

Wet. Amsterdam, i., 1892, p. 97. 
H. A. Lorentz, Over den invloed, dien de beweging der aarde op de lichtver- 

schijnselen uitoefent, Verslagen en Medcdeelingon Kon. Akad. v. Wet. 

Amsterdam, (3) ii., 1886, p. 297. 
H. A. Lorentz, De ^influence du mouvement de la terre sur lea phenomenes 

lumineux, Archives Neerlandaiscs des Sciences, xxi., 1887, p. 103. 
H. A. Lorentz, De aberratietheorie ran Stokes in de onderstelling van een aether, 

die niet overal deselfde dichtheid heeft, Zittingsverslagen Akad. Amsterdam, 

vii., 1899, p. 528. 
R. Reiff , Die Fortpflanzung des Lichtes in bewegten Medien nach der elektrischen 

Lichttheorie, Ann. d. Phys. u. Chem. (3) 1., 1893, p. 361. 
A. A. Michelson, The Relative Motion of the Earth and the Luminiferous Aether, 

Amer. Journ. of Science (3), xxii., 1881, p. 120. 
A. A. Michelson and E. W. Morley, Influence of Motion of the Medium on the 

Velocity of Light, ibid. (3), xxxi., 1886, p. 377. 
A. A. Michelson and E. W. Morley, The Relative Motion of the Earth and the 

Luminiferous Aether, ibid. (3), xxxiv., 1887, p. 333. 
A. A. Michelson and E. W. Morley, On the Relative Motion of the Earth and the 

Luminiferous Aether, Phil. Mag. (4), vol. xxiv., 1887, p. 449. 
H. A. Lorentz, De relatieve beweging van de aarde en den aether, Zittingsverslagen 

Akad. Amsterdam, i., 1892, p. 74. 
J. MacCullagh, An Essay toward a Dynamical Theory of Crystalline Reflexion 

and Refraction, 1839. 
W. Thomson, Motion of a Viscous Liquid, etc., Math, and Phys. Papers, London, 

1890, iii. art. 99. 
W. Thomson, On the Reflexion and Refraction of Light, Phil. Mag. (4), vol. xxvi., 

1888, pp. 414, 500. 

G. Green, On the Laws of Reflexion and Refraction of Light, Cambridge Trans- 
actions, vi., 1838, p. 400. 
J. Larmor, A Dynamical Theory of the Electric and Luminiferous Medium, 

London Trans., A, clxxxv., 1894, p. 719; clxxxvi., 1895, p. 695; cxc., 

1897, p. 205. 

J. Larmor, Aether and Matter, Cambridge, 1900. 
R. Reiff, Elastizitdt und Elektrizitdt, Freiburg and Leipzig, 1891. 
A. Sommerfeld, Mechanische Darstettung der electromagnetischen Erscheinungen 

in ruhenden Korpern, Ann. d. Phys. u. Chem. (3), xlvi., 1892, p. 139. 
L. Boltzmann, Vber ein Medium, dessen mechanische Eigenschaften auf die 

von Maxwell fur den Electromagnetismus aufgestellten Oleichungen fuhren, 

ibid. (3), xlviii., 1893, p. 78. 
W. Voigt, Vber Medien ohne innere Krdfte und uber eine durch sie gelieferte 

Deutung der Maxwell- Hertzschen Oleichungen, ibid. (3), Iii., 1894, p. 665. 

70 



SOUECES AND REFERENCES 71 

L. Graetz, Vber eine mechanische Darstellung der dectrischen und magnetischen 

Erscheinungen in ruhenden Korpern, ibid. (3), v., 1901, p. 375. 
O. Lodge, Modern Views of Electricity, London, 1889. 
W. Thomson, On a Gyrostatic Constitution for Ether, Math, and Phya. Papers, 

1890, vol. iii. art. 100. 
W. Thomson, On the Propagation of Laminar Motion through a turbulently moving 

Inviscid Liquid, Phil. Mag. (4), vol. xxiv., 1887, p. 342. 
V. Bjerknes, Vorlesungen uber hydrodynamische Femkrdfte, bearbeitet nach 

C. A. Bjerknes' Theorie, Leipzig, 1900. 

K. Pearson, Ether Squirts, Amer. Journ. of Math., vol. xiii., 1891, p. 309. 
A. Korn, Eine Theorie der Gravitation und der electrischen Erscheinungen auf 

Orundlage der Hydrodynamik, Berlin, 1898. 



KINETICAL PROBLEMS 
(1911-1912) 



INTRODUCTION 

IN these lectures some self-contained questions concerning kinetic 
theories are treated. They belong partly to the domain of the 
kinetic theory of gases and partly to that of the electron theory. 
Their subject was suggested by Knudsen's investigations on very 
rarefied gases and by Richardson's researches on thermionic 
currents. 

The gases offer two extreme cases which can be treated with 
comparative ease, one, in which the dimensions of the containing 
vessel are very large, and another in which these are very small 
compared with the mean free path of the molecules. 

In the former case there will be no sliding of the gas along a 
solid wall and this will have the same temperature as the con- 
tiguous gas layer, while in the latter case the gas is so rarefied 
that the molecular collisions can be disregarded. A volume- 
element does then no more contain the same matter during 
a certain time, as can be assumed to be the case for gases of 
large density. The investigation of cases falling between these 
two extremes offers considerable difficulties. In dealing with 
Knudsen's investigations one can start from the second 
extreme case, and this is the plan which will be here adopted. 



75 



CHAPTER I 

INNER FRICTION AND SLIDING, TREATED HYDRODYNAMICALLY 

1. HYDRODYNAMICAL EQUATIONS OP AN INCOMPRESSIBLE 
Viscous LIQUID 

To begin with, we consider an incompressible viscous liquid and 
write down its hydrodynamical equations. In addition to inner 
friction the sliding along a fixed wall will also be taken into 
account.* 

Let u, v, w be the velocity components of the liquid, p the 
pressure, X x , X y , etc., the inner stress, including the pressure, 
further, p the viscosity coefficient and, finally, p the density. 

For the sake of clearness it may be mentioned that X x is the 
-XT-component of the tension exerted upon a surface-element 
normal to the JST-axis, Y z the Y-component of the stress upon a 
surface-element perpendicular to the Z-axis, and so on. 

In absence of external forces the equations of motion are 

~Y +*r + ;T = (equation of continuity), . . (1) 



du du du du\ dX x 3 A,/ dX z 
-nT + w 3~ +v~-+w~-)=--~-2+ - -+ - 
vt ox oy oz/ ox oy oz 

dv dv dv\ dY x dY v c)Y z 



~:+U5-+V~ +^0-)=^ + -o + *- 

dt dx dy dz/ dx ay dz 

c>Z 



(2) 



* Helmholtz and Piotrowski investigated whether there is sliding of a liquid 
or not. They observed the oscillations of a hollow metallic sphere filled with 
liquid and suspended on a twisted wire, and they found that the sliding at the 
metal wall was not nil. Wiener Sitzungsber. zL, Abt. I., I860, p. 607 

77 



78 



KINETICAL PROBLEMS 



(3) 



- ~ 
ty 



* In deriving these formulae one considers a parallelepipcdon of edges dx, 
dy, dz within the liquid. Multiplying both sides of each of the equations (2) 
by dxdydz, we have on the left hand the product of an element of mass into 
its acceleration, and on the right hand the force acting upon this liquid mass. 

With regard to the equations (3) arid (4) we may notice that in absence of 
friction X x = Y V = Z Z = ~p, while the tangential stress components are all nil. 

In the presence of friction all stress components will be determined by ex- 
pressions depending on the manner how the velocity varies from point to point. 
The equations representing these connections will be linear ; to a first approxi- 
mation the stress components will bo determined by the derivatives of the 
velocities with respect to the co-ordinates. 

We can thus write, in general, 

X x =- 



v i ^ '^ /, *' w i. tM> 

, cx dy (-y L>Z 

and so on. 

To determine further these coefficients we take account of the symmetry 
relations. Imagine the liquid reflected at the plane x0 and notice that for 
the image the same equations must hold. It will then become plain that terms 
having an odd number of references to the JT-axis change their sign through 
the reflection, while those with an even number of such references retain their 
sign. The reference to the -X-axis can occur in a fourfold way, to wit, as in 
ox, cto, X v and Z a . 

Thus, using a reflection at the plane xQ, we shall find that X x remains 
unchanged, and this gives a, 2 a 21 --a 13 =a sl -0. Again, Y g remains what it 
was, giving 6 12 = 6 21 = & 13 = b n = 0. 

Next, a reflection at the plane y=Q leaves X r unchanged, whence 23 = a 32 = 0, 
while Y x changes its sign, and this gives 6 n = 6 22 = 6 33 = 0. 

From the fact that the exchange of the Y- with the Z-direction can have 
no effect upon X x , it follows that 22 =a 33 . 

The formula for X x thus becomes 



which, in view of the equation of continuity, can be written 



Taking into account that the couple resulting from the tensions upon an 



1 INNER FRICTION AND SLIDING 79 

With regard to the boundary conditions we may notice that, 
if there be no sliding, the velocity of the liquid in contact with a 
solid body is the same as the velocity of the latter, but if there is 
sliding, this equality holds only for the normal velocity com- 
ponents. In the latter case, therefore, the relative velocity of 
the outermost layer of the liquid and the solid wall has a tangential 
direction. 

Let us now consider within the liquid near the solid wall a 
short cylinder whose dimensions along the normal of the wall 
are infinitely small compared with those in tangential directions, 
and let us express the condition that the forces exerted by the 
liquid upon this volume-element are in equilibrium with those 
due to the solid body. 

Let h be an arbitrary direction in the tangential plane. Then 
the force exerted by the liquid, per unit area, in the direction h 
may be represented by H H9 and that exerted by the solid body by 
\v h , where v h is the relative velocity of the liquid and the solid 
wall and X a proportionality factor. Thus the boundary condition 
will become H M -Atfc (5) 

If there be no sliding, then v h =0 and we must put X=oo. 

Let the state of motion be stationary, so that Bu/ffc, vvfit, 
dw/dt all vanish, and let us further assume that the velocities are 
so small that their products and their derivatives can be neglected. 
Then, in virtue of (1), the equations of motion (2) become 



- 



. (6) 



element of liquid must be zero, if we limit ourselves to magnitudes of the third 
order, we find Y n = Z y , whence 6 23 = 6 3a . Thus, 



Again, since the liquid is isotropic, the expressions for Y, and Z % must 
follow from that for X x by a cyclic permutation, and similarly those for Z 9 and 
X v from that for 7 r Whence it follows that the coefficients in the formulae 
for X x , Y v , and Z, must be equal, and similarly for Y t , Z x , and X,. 

The isotropy of the liquid implies also that if the equations be transformed 
to a new system of axes obtained by a rotation of the original one, the co- 
efficients retain their values. This gives a n - a >9 = 26 M = 2/*. 



80 KINETICAL PROBLEMS , OIIAP. 

2. EFFECT OF SLIDING UPON A LIQUID FLOWING IN A TUBE 

As a first application we consider a liquid flowing through a 
narrow tube. We put the X-axis along the tube. 

The equations of motion can be satisfied by putting v = w =0. 
This amounts to disregarding small lateral motions near the ends. 
The equation of continuity (1) then calls for 3w/3a;=0, so that 
u and therefore also Aw become independent of x, and, by the first 
of (6), the same is true of dp/dx. By the remaining two equa- 
tions (6) the pressure must also be the same all over the cross- 
section. This will also be the case of dp/fix. Consequently the 
pressure varies uniformly along the tube and depends but on a 
single co-ordinate, x. If p l and p 2 be the values of p at the 
beginning and the end of the tube, and if I be its length, we have 

^=2-^1 

dx "" "/ 

and, by (6), 

Vu Vuj^Vi 

ty* dz z pi ' 

Limiting ourselves to a tube of circular section, of radius /?, 
transforming to polar co-ordinates, and noticing that, since the 
motion can be assumed to be axially symmetrical, u depends 
only on r, we can write the last equation 



d*u 1 du^pt- 
dr 2 r dr p.1 



whence, by integration, 



an additive integration constant being omitted, since the velocity 
of flow is a maximum at the middle of the tube, so that du/dr=0 
for r =0. Integrating once more, we find 



where C is to' be determined from the boundary condition. In 
absence of sliding u - f or r = fi, whence C = - jR 2 , and therefore, 



i INNER FRICTION AND SLIDING 81 

The amount (volume) of liquid streaming through a cross- 

f* 
section is, per second, 27r; urdr. Thus, in our case, 



) 



which is proportional to 1Z 4 , in agreement with Poiseuille's law. 

In the presence of sliding the friction force -pdu/dr is equal 
and opposite to the force exerted upon the liquid by the tube 
walls. The boundary condition then becomes 

du * t D 

-ft, ,- =AU, for r =JK, 

whence 



and 



The corresponding amount of liquid streaming across the section 
of the tube is, per second, 



The term 4/i/XU gives the correction to Poiseuille's law for the 
sliding. For X = oo, or no sliding, (8) reduces to (7). 

3. THE DRAGGING OF A LIQUID BY A MOVING PLATE 

For a second illustration of the hydrodynamical equations let 
us take the case of a liquid contained between two flat plates 
of infinite extension, perpendicular to the Y-axis. Let the 
lower plate (y =0) be at rest, while the upper plate (y = A) moves 
uniformly with the velocity a along the ^-axis. Since the liquid 
is dragged by the upper plate, so that its velocity increases along 
the Y-axis, we will write for the velocity components 



The friction upon any plane parallel to the plates is, per unit 
area, 

du 

*q-i*>* 

VOL. I G 



82 KINETICAL PROBLEMS CHAP. 

For y=0 we have u=^C l9 so that the relative velocity of the 
liquid and the lower plate is C l9 and therefore the force exerted 
by the latter upon the liquid \C V Thus the first boundary 
condition becomes 

du *~ 

/Av-=AC 1 . 

y 

This gives ' 



For y = A, u = Cj + C 2 A, so that the relative velocity of the liquid 
and the upper plate is a-(C f 1 + C f 2 ^)> an d the force exerted by 
this plate upon the liquid \(a -C 1 - C 2 A). The friction of the 
liquid against this plate is -/zC^ an( i the second boundary 
condition thus becomes 



or, substituting the value of C l9 

Xa 



whence the friction 



For X = oo we have C 2 =a/A, so that for finite X the friction will 
be somewhat smaller. 

Notice that /*/X has the dimensions of a length. (This follows 
from the equation fjLdu/dy=\C l9 since C l is a velocity, to wit, 
C l =u for y = 0.) The physical meaning of this length can be 
seen by imagining that either plate is moved away from the 
other over the distance /4//\, while the liquid expands so as still to 
extend from plate to plate and its state of motion remains un- 
changed. The velocity of the liquid relatively to either plate 
is then nil at both boundaries. In fact, w=0 for y = -/x/X, and 
w=a for y = A+/A/X. Thus the solution of the problem with 
sliding can be reduced to that of the problem without sliding, 
provided the liquid is given the said expansion. 

Also the result of the problem of a liquid flowing through a 
tube of circular section is in harmony with this property. For, 
if R in the expression (7) is replaced by JR+/i/X, the expression 
(8) follows, provided /u/X is small in comparison with R so that 
the square of /tt/XJK can be neglected. 



INNER FRICTION AND SLIDING 83 



i. EFFECT OF SLIDING IN THE CASE OF TRANSLATIONAL 
MOTION OF A SPHERE IN A LIQUID 

We will now determine the resistance offered by a liquid to 
a sphere endowed with uniform rectilinear motion. This will 
lead us to the famous formula of Stokes which, among other 
things, comes into play in the lately developed theory of Brownian 
movement. It is still an open question how far the validity of 
this formula can be upheld for very small particles and irregular 
motions. This will still be discussed in the sequel (Art. 7). 

In dealing with the problem in hand we will assume that not 
the liquid as a whole is at rest and a sphere moves through it 
but, inverting the relations, we will imagine that the sphere is 
at rest and the liquid moves past it, having at infinity a uniform 
rectilinear motion. We put the Z-axis along the direction of 
this motion and take the centre of the sphere as the origin of 
our co-ordinate system. Thus, if u, v, w be the velocity com- 
ponents of the liquid, we have at infinity w=0, v = 0, w-a. 

From our previous equations (6) 



and from the equation of continuity (1) we derive 



To solve these equations we will follow Kirchhoff and intro- 
duce an auxiliary function <E>, such that 



To begin with, we could try as a solution 

ao ao ao 

w = , 0=3-, w= ~ . 
dx dy dz 

This satisfies the equations of motion, but not the equation of 
continuity. In fact, since 

du dv dw 
x- + ~-+^- 

dx dy dz 

the equation of continuity could only be satisfied if 



84 KINETICAL PROBLEMS CHAP. 

i.e. if the pressure were everywhere nil. We try, therefore, to 
help matters by adding new terms and write 



, , , 

u=*~- +u,v=*-+v 9 w= ~- + w. . . (9) 
fix fiy fiz ^ ' 

If the new terms can be so chosen that 

Aw'=0, A*/=0, Aw'=0 . . . (10) 
and 

?)?/ fiv' fiw' A ~ v /11X 

+ - + =-A<D=-A . . . (11) 
ex vy cz \t, 

our solution will be ready. Now, noticing that we must have 
Ap = and that this is satisfied by spherical harmonics of which 
the simplest are l/r and all partial derivatives of 1 /r, we try to put 



Account is here taken of the circumstance that the pressure 
p must be an odd function of z, since it must have different 
signs at the opposite poles of the sphere. Further, the function 
of z chosen is the simplest which satisfies Ay=0. Finally, a 
constant factor is inserted which will presently be given a 
suitable value. An additive constant would for the problem in 
hand be without significance, and is therefore omitted. 

Equations (10) and (11) are now satisfied by 



r 

Next, we put 

fi /1\ fir 

(- } + ttr-. 

i\rj vz 
The first two terms do not contribute to A4>, and since 



the last term gives 



Notice that all three terms of 4>, though odd functions of z, 
are even functions of x and y, which harmonises with the 
symmetry of the liquid motion. 



i INNER FRICTION AND SLIDING 85 

The first term gives at infinity, where the derivatives of the 
remaining two vanish, 

w-<> 
w -~-=a. 

vz 

Thus is a introduced into the formula. The middle term enables 
us to satisfy the boundary conditions at the surface of the 
sphere. By means of these conditions the constants b and c can 
be expressed in terms of the velocity a and the radius R. 
Thus the solution becomes 



36 



w = a + 



. (12) 



Thus far we have followed Kirchhoff. In the boundary conditions 
at the surface of the sphere we will take account of the sliding 
and in that deviate from Kirchhoff. Since at the surface of the 
sphere the velocity of the liquid must be tangential, the first 
boundary condition is 



whence 



Equations (12) show that there is symmetry around the 
Z-axis. Let us then consider a point P at the surface of the 
sphere in the plane XZ } for which POZ = 0, and let us introduce 
a new orthogonal system of axes, of which OZ' passes through 
P and OX' lies in the plane XZ. The equations (12), when 
transformed to the new axes, become 

u' .= - (a - -- jsh 1 6 + ( -JT - -r )x'(z' cos 6 -x' sin 6) 
\ f f/ \r* f d / 

v' = (^ - )/(' cos fl - *' sin 6) . (14) 



-* sn 



86 KINETICAL PROBLEMS CHAT. 

The velocity of sliding at P follows from these equations for 
x 9 =y' =0, z' =R. Thus, and by (13), its components are 



u'- - a -i-& sin > f-0 f tf-0. 

Whence, the tangential tension at P exerted by the sphere upon 
the liquid, 



and the friction component, which must be in equilibrium with 
this force, 



Thus the boundary condition becomes 

6 c\ . 



where z' =fi and x' =0. 

Evaluating the left-hand member by means of (14), one finds 



f = - -4 sin , 
and the second boundary condition assumes the form 



Since has disappeared, this boundary condition can at once be 
satisfied all over the surface of the sphere. 
Formulae (13) and (15) give 

i , W 



Thus b and c are expressed in terms of known data, and sub- 
stituting these values into (12), we find also u, v, w. 

The total force exerted on the sphere by the liquid will fall, 
by reasons of symmetry, into the Z-axis and can thus be found 
by integrating Z z < over the surface of the sphere. Now, Z z > can 
be determined, in two ways, either from 



i INNER FRICTION AND SLIDING 87 

or from the tangential force X t . and the normal force Z z , 
leading to 



To choose the latter way, Z^ must be evaluated from the 
equation 

Z*' = -p f 2p. , for x' =0, z' = B, 

which gives 

' 6/xc ,, 12u6 * 
Z z > = 2 cos 6 - -jg cos 0, 

and since we have already found 

X z > = - -- sin 6, 
we have 



whence, by integration over the surface of the sphere, the required 
resistance, 

W = 2irR 2 l Z z > sin 
i.e. 



+ A72 
For X == oo this expression reduces to 



which is Stokes* well-known formula. 

Such, then, is the resistance experienced by a sphere moving 
with velocity a through a stagnant liquid. 



5. LIQUID MOTION DUE TO AN IMMERSED VIBRATING PLATE 

We shall next investigate how far this resistance formula 
can be assumed to hold for a sphere in non-uniform motion, 
such as the Brownian movement. The problem offers con- 
siderable difficulties, and we shall, therefore, confine ourselves 



88 KINETICAL PROBLEMS CHAP. 

to the simple case in which the body has a vibratory motion of 
translation. 

As an introduction we will consider a flat plate in the 
FZ-plane maintained in non-damped vibration in the direction 
of the"Z-axis. Let the plate be unlimited, so that the state 
will be the same all along the Y- and the Z-axes and thus 
depend on x and t only. The velocities being again assumed to 
be infinitesimal, and all relevant magnitudes being functions of 
x and t only, the equation of motion 

(dw div f)w dw\ r)n A 
P\ n.-+w~ - +^r f w~- )= -' +LL&W 
r \dt dx (it/ dz/ dz ^ 

reduces to 



As its solution we take 

w = 

Substituting this into (16), we have 



*, ..... (17) 
whence 



Thus, 

ui 

of which the real part is 



representing waves which issue from the plate along the positive 
-X-axis, this result being obtained by taking for /3 as solution of (17) 
the negative root. The wave-length is 27r\/2///wp. That this 
has the dimensions of a length is manifest by (16). The amplitude 
of the oscillations of the liquid is thus decreasing considerably 
when these are propagated over a wave-length, namely, to e"" 2ir 
times its original value. 



i INNER FRICTION AND SLIDING 89 

6. EFFECT OF FREQUENCY IN THE CASE OF A SPHERE 
VIBRATING IN A LlQUID 

Passing to the case of a sphere vibrating in the direction of 
the Z-axis, we may represent its velocity component w by ae int , 
where n is real, corresponding to non-damped vibrations. The 
motion of the liquid must satisfy the equations of motion 



and the equation of continuity 

Su dv_ ^_A 

fix dy ?z ' 

while at the surface of the sphere, if sliding be disregarded, 
w = v = 0, w=ae int . 

Of these equations two particular solutions can be found, 
whose superposition gives a solution which satisfies the boundary 
conditions at the surface of the sphere. 

For the first solution we put p=0. This will then exhibit 
some similarity with the problem of the vibrating plate, since in 
the case of the latter the equation of motion (16) was free of terms 
containing p. 

Let us, therefore, introduce an auxiliary function <f> satisfying 
the equation 



and depending on t and r only. The equations of motion will be 
satisfied not only by <I> itself but by its derivatives as well, and 
we can put 



Of these the last is so chosen as to satisfy the equation of 
continuity. 

The differential equation for 4> becomes, in polar co-ordinates, 

WrO) 
"- 



this being of exactly the same form as the equation of motion 



90 KINETICAL PROBLEMS CHAP. 

(16) of the liquid containing a vibrating plate. The solution is 
thus 



where /9 - - /^(l +0 and 6 is a constant which for the present 
>r 2/A 

may be left undetermined. Thus far the first solution. 

The peculiarity of the second solution is that it annihilates 
the last terms in the right-hand members of the equations of 
motion. As such we introduce a function ^ of 2 and r which 
satisfies the equation 



If then u, v, w are equalled to ^ or to any of its derivatives, 
we shall have A u = A v = Aw = 0. 

In order to satisfy the equations of motion and the equation 
of continuity as well we put 



T"T> T> 

dydz r dz* ^ 

and we take for ^ the simplest harmonic function f ) multiplied 

by the factor e ini , to express non-damped vibrations, and by the 
density p, to do justice to the equations of motion ; in fine, 



where the constant c is again left undetermined for the present. 
Thus the general solution of our equations becomes 



dxdz p dxdz' 



v =--3- +- - % , 
oyoz p oyoz 



The amplitudes of both functions can be compounded by putting 

, 6 ., c 



i INNER FRICTION AND SLIDING 91 

Next, if we introduce the amplitudes u', v f , w', defined by 
tt=tt'e tnt , etc., these will satisfy the equations 



-, , 

vyoz \vx 2 oy 2 / 

The boundary conditions at the surface of the sphere are, in 
absence of sliding, u' =0, v' =0, w' =a, for r =R. 
The first of these gives 

V-r^-O 
dr V~ U ' 

whence 

0. . . . (18) 



The second, v'=0, gives the same equation, while the third 
boundary condition leads to 



. _ 

~fdr r* dr 'r* dr* > 

or, since dfldr-rd i f/dr*=0, 

2df , 
-rdr =a ' ioTT=R ' 
whence 

. . (19) 



From (18) and (19) we have, for b and c, 

.... (20) 
. . . . (21) 



Thus the state of motion is determined. 

In view of the symmetry with respect to the Z-axis the 
resistance opposed by the liquid to the motion of the sphere can 
be represented by 

27rB 2 f Z n sin OdO, 
Jo 

where 6 is the angle between the normal n (of the sphere) at a 
point of the XZ-plane and the Z-axis. 



92 KINETICAL PROBLEMS CHAP. 

If we put Z n =Z n <P*, Z m -tijP*, etc., then 



Evaluating Z n by means of the condition df/dr-rd 2 f/dr 2 =Q, for 
r = R 9 the equations (20), (21), and the relation (17), we find for 
the required resistance, K = 2'rrR 2 f ir Z n sin 0d0 9 

K = MR( - 1 + IfiR - IpR^aJ*". . . (22) 



For very slow vibrations, i.e. for very small values of }R, this 
reduces to Stokes' formula 



For any frequency of vibrations (22) can be written 



The velocity of the sphere will then be 

w = a cos nt y 
and the force opposed to the sphere by the liquid, 

K =/Lt0a cos nt -/xAa sin nt. (23) 



Only the first term is to be considered as a resistance, if " resist- 
ance " be so characterised that the work done by it over a 
complete period is negative. Now, the work of K is 

fKwdt =f(pg(i 2 cos 2 nt -p,ha 2 sin nt cos nf)dt y 



and here the first term, whose coefficient g will presently appear 
to be negative, gives on integration over a complete period a 
negative quantity, whereas the second term gives nothing. The 
term - fJia sin nt is a force proportional to the acceleration ; the 
corresponding effect is thus an apparent increase of the mass of 
the sphere, which is due to the co-vibration of the liquid. 



i INNER FRICTION AND SLIDING -93 

Let us still consider the resistance pga cos nt. Here g is the 
real part of 

and since 



we have 



and the resistance becomes, with w s written for the velocity 
of the sphere, 

(24) 



+R I 



This is again the resistance according to Stokes, but increased 
by a term which for high frequencies can outweigh the Stokes 
resistance. 



7. THE QUESTION OF VALIDITY OF STOKES' LAW FOR 
BROWNIAN MOVEMENT 

We can now ask what this resistance is like when the sphere 
is endowed with any variable motion. This is of importance 
in connection with Brownian movement. But we must limit 
ourselves to small velocities, so as to be able to neglect, as above, 
all terms such as udufix, etc. If the velocity were known as a 
function of the time for the whole duration of the experiment, 
it could be developed into a Fourier series, and the result (24) 
could be applied to each term. Thus the resistance would 
be found. This, however, is not very helpful for a general 
discussion. 

Stokes' formula cannot be applied to the case of Brownian 
movement, as this is much too quickly variable for such a 

purpose. The term R n P. in (24) is to be neglected in presence 

V 2//, 

of unity, if the vibration time T is large compared with 7rpR 2 lfjL. 
If we write TrpR 2 //* = 0, then it will be possible to apply Stokes' 
law for slow vibrations, for which, that is, the vibration time T 
is large in comparison with 6, and this holds also for other 



94 KINETICAL PROBLEMS CHAP. 

motions, provided the velocity does not change much during the 
time 9. 

To illustrate this by an example, let us see whether Stokes' 
law is applicable to the extinction of the motion of a sphere due 
to the friction of the liquid. If m be the mass and v the velocity 
of the sphere, Stokes' law would give 

dv 

m dt ~ 
whence 



so that v would dwindle down to the e-th part of V Q after the time 
w/67r//,R. If p t be the density of the sphere, this time is 



If p and p l are comparable with each other, 6 is seen to be of 
the same order as the time r. Stokes' law cannot, therefore, be 
applied to the extinction of motion here considered. 

For particles in Brownian movement becomes quite small. 
If # = 5 . 10- 5 and ,4 = 18 . 10~ 5 , then 

a 7T.25.10- 10 A 1A . 

u sss ' ' A x 1 o ~ 4 J.U p. 
i Q in-5 * ' 

lo . lu 

approximately. Yet the motion of the suspended particles will 
vary during this time considerably, so that the refinements of 
the Brownian movement cannot be mastered by the law of 
Stokes. In many other cases, however, the motion within the 
time will change but little and the law will be applicable. 

8. DEDUCTION OF EINSTEIN'S FORMULA FOR THE MEAN SQUARE 
OF THE DEVIATION OF A PARTICLE IN BROWNIAN MOVEMENT 

For the investigation of Brownian movement it is of import- 
ance to correlate the mean square 2 of the distance attained by 
a particle within a given time t with the properties of the liquid, 
i.e. with the viscosity coefficient p,, and with the radius R of the 
particle. 

Without following the particle's actual crinkly path and its 
rapidly variable motion, we can find the required connection by 



i INNER FRICTION AND SLIDING 95 

a roundabout way, namely, by considering the diffusion velocity 
of the particles. On the one hand this can be expressed by /a, 
and on the other hand by *. Evaluating thus the diffusion 
coefficient by two different methods and equating the two 
expressions, we shall find Einstein's formula * for f a . 

First method. Suppose we had a liquid containing suspended 
particles whose concentration varies in some direction or other. 
Along that direction put the Jf-axis. The concentration of the 
particles is given by their number n per unit volume. Thus n is 
a function of x. The particles will exert an osmotic pressure. 
Now, if it be assumed that the mean kinetic energy of a particle 
is equal to that of a gas molecule at the same temperature T, 
i.e. $kT, the osmotic pressure is 

p=nkT, 

this being f of the total kinetic energy of the particles per unit 
volume. 

The force driving the particles is equal to the difference of 
their osmotic pressure in two planes perpendicular to the sc-axis. 
In a stationary state of diffusion this difference will be balanced 
by the force exerted by the liquid upon the particles.f 

Apart from the Brownian movement the particles have a 
common velocity along the X-axis if the concentration decreases 
with increasing x. 

Since the formulae for the stress components and the equations 
of motion of the liquid are linear (products of velocities and 
derivatives of velocities being omitted, as they are small), we 
may say that the force exerted by the liquid upon the particles 
consists of two parts, one corresponding to their common velocity, 
and another to their Brownian movement. The latter will vanish 
for all the particles taken together. 

Thus we have , 

-~ = 67r^Rnv 9 

* Ann. der Physik, vol. xix., 1906, p. 371. 

f The osmotic pressure is, properly speaking, the momentum along the 
X-axis transferred by the moving particles in the positive less than in the 
negative direction, per unit area perpendicular to this axis and per unit time. 
Consequently, the momentum of particles contained in a layer of thickness dx 

undergoes the change - -^dx per unit time, i.e. as if a force of this magnitude 
acted upon the particles. 



96 KINETICAL PROBLEMS CHAP. 

where the left-hand member represents the gradient of the 
osmotic pressure, and the right-hand member the resistance 
opposed by the liquid to n particles ; and since p=nkT, 



Whence the diffusion current 

dn 

w -K-f-9 
ax 

where K, the coefficient of diffusion, has the value 



The diffusion is thus calculated by means of the law of Stokes. 

In the second method we concentrate all our attention upon 
the Brownian movement. We do not consider, however, the 
actual path of a particle, but its total displacement within a 
certain time. Let the distance attained after a time t, reckoned 
from some initial moment, be s. Then s is different for different 
particles. By means of probability considerations it can be 
shown that the mean value of s 2 is, for all particles, proportional 
to t, say, 

s 2 =#. 

The coefficient ^8 can be observed, as, among others, was done 
by Perrin.* 

Thereupon will our diffusion theory be based. Suppose that 
the concentration varies from point to point ; then the particles 
from a small volumfe-element will spread after a time t over a 
sphere of radius Vfit. Thus the concentration differences will be 
gradually obliterated. 

Now, it can be shown that 

-# ..... 

where f is the projection of s upon the -X-axis.f 

* Comptes rendus, Paris, vol. cxlix., 1909. p. 477, and vol. clii., 191 1, p. 1569. 

f Suppose that during the time t all particles are displaced along the X-axis 
over the same distance Z, so that for one half of the particles x is increased, 
and for the remaining half diminished by 1. Next, consider a plane V, per- 
pendicular to the JL -axis, and two layers, each of thickness I, on both sides of V. 
Let N t be the number of particles contained, at the beginning of the time t 



i INNER FRICTION AND SLIDING 97 

From (25) and (26) follows Einstein's formula 



where R is the gas constant and N the number of molecules per 
gram molecule. 

By means of this formula Perrin determined N from his 
experiments. 

and por unit area of V, in the layer on the positive side, and similarly N 2 
on the negative side of V. Then the diffusion per unit time is |(2V 2 - NJ/t. 
We can assume that within these thin layers n is a linear function of x, and shall 

thus find, for the diffusion, - ~ --- , and, for the coefficient of diffusion, 

U a 



which obviously must be replaced by (20), when account is taken of the 
diversity of the ^-values for different particles. 
It will be readily scon that * = &*. 



CHAPTER II 
FRICTION AND SLIDING, TREATED KINETICALLY 

9. FRICTION INDEPENDENT OF THE DENSITY OF THE GAS 

KETURNING to the questions concerning inner friction and sliding, 
we will now treat them on the kinetic theory. Moreover, instead 
of a liquid, we shall now consider a gas. ' If this be strongly 
condensed, there is no sliding. Thus we can at first exclude the 
sliding, to introduce it later on as a correction. 

We will begin by proving that the inner friction is independent 
of the gas density. For this purpose we consider a simple case 
of motion, viz. a gas whose horizontal layers are, as a whole, 
shifted 'over each other. Let us introduce a co-ordinate system 
whose -XZ-plane is parallel to these layers, the X-axis pointing 
in the direction of streaming. If u be the velocity of a layer, 
the motion of the gas can be expressed by the equation 



where c is a constant, it being assumed that the layer y Q is 
at rest. 

The state of a molecule will be determined by f, ?/, *, the 
velocity components, and x, y, z, the co-ordinates of its centre 
of gravity. The relative co-ordinates and velocities of the parts 
of a molecule with respect to its centre of gravity can here be 
disregarded, as they do not affect our problem. Let us consider 
the molecules which at the instant t are contained in the volume- 
element dS=*dxdydz at the point P, of co-ordinates x, y, z, and 
whose velocity components are contained between f and g + dg, 
t) and rj+drj, % and ?+rff. The number of these molecules can 
be expressed by ,, { . ,, 

08 



CHAP, ii FRICTION AND SLIDING 99 

where d\=d^drjd^ and F is independent of x and z, since, by 
assumption, the state is the same throughout a gas layer parallel 
to the -XZ-plane. 

Further, let bdSd\dt be the number of molecules which are 
thrown into this group by collisions during the time dt, and 
adSd\dt the number of those which for the same reason and 
during the same time leave this group. Let external forces be 
absent. At the time t + dt the molecules of this group will come 
to lie in a volume-element dS' =dS [by Liouville's theorem], at 
the point x + %dt, y + rjdt, z + tdt. Thus, 



f(f, *), , y> WSdX + (6 - a)dSdXdt 

will be the number of molecules contained at the instant t + dt 
in the element dS, constructed at the point x + ^dt, y + ydt, 
z + fcfe, and having their velocity components within the domain 
d\. Whence, ' % f ^ f 

b -a =^-77 + -,-. 
cy ' M 

This equation would enable us in general to find the properties 
of the function F. This, however, can be accomplished only in 
the simplest case. If we put J=/for y=0, then 



If the whole gas is given a translation velocity -cy along the 
#-axis, the state of motion in a plane at the height y becomes 
what it was originally in the plane j/ = 0. Whence we see that, 
generally, 



We assume further the state to be stationary, so that 

M 1 n , dF 
^-=0, 6-a= - rj. 

dt Cy ' 

It remains only to determine the state for j/=0. From the 
preceding equation we have, for y =0, 

6-a--0|4 ..... (28) 

We will now compare the state of the gas in the said motion with 
the state in which there is no streaming, i.e. for c=0. For this 
case / can be written Nf Q (where N is the number of molecules 



100 KINETICAL PROBLEMS CHAP. 

per unit volume). Then, in the state of motion, f^N/Q+f, 
where the function /' contains all the refinement; it may 
assume positive values for some groups, and negative for 
others. The exact determination of /' is laborious and for 
our purposes unnecessary, since it can be shown by a simple 
reasoning that the friction is independent of the density. We 
assume that the velocity gradient c is small, in other words, 
that there is only a small departure from the state of rest, so 
that /' is small compared with N/* . Then / in the right-hand 
member of (28) can be replaced by Nf Q , which amounts to 
neglecting only terms of a higher order. This gives 



For the state of rest Maxwell's velocity distribution * holds, and 
&-a = 0. Owing to the change brought about in the function 
/ by the motion b - a does no longer vanish. We consider again 
a certain group of molecules and we distribute also the remaining 
molecules into groups according to their velocities and co- 
ordinates. Let the numbers of molecules in these groups be n l9 n z , 
etc. Owing to the collision of a molecule of the first group with 
a molecule of one of the remaining groups that molecule leaves 
its group. Such a collision then belongs to the type a. Collisions 
of molecules of the remaining groups with each other can increase 
the numerosity of the first group and will thus be of type 6. 
Since the number of collisions between two groups of molecules 
is proportional to the numerosity of the first as well as to that 
of the second, we can write 



where a 12 is positive for such collisions as contribute something 
to 6, and negative for such as contribute to a. 

For c=0 the numbers of molecules in these groups will be 
denoted by n 10 , n^, etc., so that 6-a = 

In general we can put 

6 - a 



* Viz - : flt . 

/.(fc* ft 
where h is inversely proportional to the absolute temperature. 



n FRICTION AND SLIDING 101 

where the a's remain as before, since we are considering groups 
which are completely determined by their velocities, so that 
when we pass from the state of rest of the gas as a whole to its 
streaming only the numerosity of each of the groups is changed. 
Since we consider only a small departure from the state of 
rest, %', n 2 ', etc., are small compared with w 10 , n^, etc., and 
therefore 

b - a = Sa 12 ( 



This leads to the equation 

(29) 



by means of which the variation of / or the numbers n/, n 2 ', 
etc., can be determined. If the molecules are divided into K 
groups, there are K equations such as (29). 

If we change c, the equation (29) remains satisfied, provided 
%', n 2 ', etc., are changed in the same ratio; thus the whole 
disturbance, and therefore also /', will be proportional to the 
velocity gradient. Now, if the density be varied, and thus also 
N, the number of molecules in each group for the state of rest, 
n 10 , w 20 , etc., varies in the same ratio. Then (29) is satisfied, 
provided %', r& 2 ', etc., remain the same, so that also/' does not 
change. 

Whence we see that /' is proportional to c and independent 
of N, that is to say, independent of the density. 

The friction upon a plane parallel to the ZZ-plane is, per 
unit area, 



In fact, this is the momentum along the -X"-axis, transferred per 
unit time across a unit area parallel to the X Z-plane. Since in 
the state of rest there is no friction, 



and the last expression for the friction can also be written 



This, as /' itself, is proportional to c and independent of N 9 and 

can, therefore, be written 

//,c, 

where /A is the coefficient of friction. The latter is thus seen to 
be independent of density. 



102 KINETICAL PROBLEMS CHAP. 



10. UNIFORMITY CONSIDERATIONS 

That friction is independent of density can also be proved 
by another method which, though not going so deep into the 
phenomena, is yet exact. We shall in this case also confine our- 
selves to a simple example, but taking at the same time account 
of the sliding. We consider two infinitely extended horizontal 
plates, of which the lower is at rest and the upper moves with a 
constant velocity v horizontally, towards the right, and we pro- 
pose to find the motion of the gas contained between the plates. 
Fig. 1 gives a graphical representation of the motion of the 
gas for the case in which there is no sliding along the walls. The 
abscissae are the velocities in the different layers, and the 
ordinates the distances from the lower plate. The velocity 
v of the upper plate is represented 

* by AB. Manifestly, the line join- 
ing the end-points of the velocity 
vectors is a straight line. 

A sliding along the walls will 
have the effect that the velocity 



^ FIG. 1. ^ ^ *ke 8 as l aver m contact with 

the upper plate will be smaller. 

If this be represented by AF (Fig. 2), then AF<AB. Similarly, 
the velocity of the layer touching the lower plate will no longer 
be nil, but will have some small value OE. The distribution 
of velocities over the different layers will no longer be represented 
by a straight line but by the line EF in Fig. 2, which, when the 
plates are far enough apart, has a straight portion, but is curved 
at the extremities. 

To enter somewhat deeper into the sliding along the walls, 
we must keep in mind that by the velocity of a gas layer is 
meant the average velocity of the particles of which the layer at 
a given instant consists. Among the particles of the lowermost 
layer there will be some which previously belonged to a layer 
with some velocity of streaming and were carried down to that 
'layer by collisions. For these ascertain direction of motion, 
viz. that of the streaming velocity, will be privileged. In this 
layer, however, there will also be present, at the given instant, 
some particles which were there before and which collided with 



II 



FRICTION AND SLIDING 



103 



the solid wall. We have, therefore, to distinguish also different 
cases of collision with the wall. If the wall is a perfect reflector, 
then the particles retain after the collision their tangential velo- 
city component. But in general even the best polished walls will 
have to be considered as rough with regard to impacts of mole- 
cules, so that the reflection will be of a diffuse nature, and there 
will be no privileged direction of motion for the rebounding 
particles. But, as we just saw, there is such a direction for the 
impinging particles, so that, all things being considered, the 
layer in contact with the wall will have a velocity in the direction 
of streaming, very much as in* the case in which there is sliding. 
The precise form of the curve representing the velocity distribu- 
tion is hard to deduce from these considerations, at least for 
points near the plate, where the state of motion is very com- 
plicated. Among 'other things one would have to take into 
account that some of the molecules colliding with the wall 
adhere to it. But the farther away from the wall, the less the 
irregularities, and if we assume the mutual distance of the plates 
large compared with the mean 
free path of the molecules, a con- 
siderable portion of the curve will 
be straight. If the distance were 
very small, the straight-line portion 
would entirely disappear. 

In Fig. 2 the velocity curve is 
represented by EF. (Probably 
its shape should be somewhat 
different, such, e.g., as indicated by 
the dotted line.) If the straight- 
line portion of the curve be pro- 
duced up to the points C and D, 

the former being on the Y-axis and the latter having the 
abscissa AB=v, then C and D will have the following signifi- 
cance. If the gas were replaced by an imaginary gas layer 
extending also beyond the plates and having throughout the 
same constant velocity gradient as that prevailing in the actual 
case only between the plates, then the layers of gas at C and 
D would have the same velocities as the plates, i.e. zero at C 
and v at D. (Cf. Art. 3, p. 82.) 

We will now prove that the segments OC and BD are 



G 



O 



C 





X 



. 2. 



104 KINETICAL PROBLEMS CHAP. 

independent of the distance of the plates. For this purpose let us 
imagine a plate P inserted at the height of G and moved in such 
a way that the state between G and A is not changed, except, 
of course, in the immediate neighbourhood of the plate P itself. 
We make the arc of the velocity curve near P congruent to that 
at E. Now if HK=OE, the velocity distribution represented by 
KF can prevail, provided GH is equal to the velocity of the plate P. 
In fact, if the whole system is given a common velocity equal 
and opposite to GH , the velocity curve retains the same shape, 
but is simply shifted over the distance GH to the left. The state 
at the plate P will then be exactly the same as that actually 
prevailing at the lowermost wall. Whence it follows that 00 
and BD are independent of the mutual distance of the plates. 

In much the same way it can be shown that OC=BD. In 
fact, if the whole system is given a velocity <fequal and opposite 
to AB, the upper plate is brought to rest, while the lower one 
will have a velocity v towards the left, and the velocity curve 
will be shifted over a distance AB in the same direction. Thus, 
if the figure is turned around by 180, it must be exactly the same 
as that representing the actual state. The shape of the curve at 
E must, therefore, coincide with that at F, and thus OE = FB 
and also OC=J5D. 

Further, if the velocities of the plates are supposed to be very 
small compared with those of the molecules, the state of the 
gas can be considered as an infinitesimal deviation from the state 
of apparent rest. This deviation can then be put proportional 
to the infinitesimal cause, i.e. to the velocity v of the upper plate. 
All horizontal lines of the figure can thus be magnified or reduced 
in the same ratio as v without changing 00 and BD. Whence 
we see that 00 and BD are independent of the velocity v. 
Again, as we saw before, OC and BD remain also unaffected by 
the change of the distance between the plates. They can thus 
depend only on the nature and the density of the gas. All this, 
of course, holds only if the velocity curve has a rectilinear portion, 
and ceases, therefore, to be true when the distance between the 
plates is of the order of the mean free path of the molecules. If 
we put OC=BD = v, then v will be characteristic for the gas. 
The meaning of v is, that if the velocity gradient were constant 
not only between but also at the walls, the gas at a distance v 
from the plate would attain the velocity of the plate itself. 



n FRICTION AND SLIDING 105 

Let the velocities of the plates be zero and v, and their mutual 
distance A. Then, by what precedes, the velocity gradient of 
the gas (i.e. the change of velocity per unit length in the direction 

of the F-axis) will be ---- , and, therefore, the friction per 

A+2j; 

unit area, parallel to the plates, -~ . 

A -\-2iv 

We will call v the coefficient of sliding and we will prove that 
it is inversely proportional to the density p, while it will be 
shown once more that ^ is independent of p. 

For this purpose let us consider a second system which will 
be denoted by II., while the original system will be denoted by 
I. ; the magnitudes relating to the system II. will be distinguished 
by dashes. Let tha system II. be so chosen that the correspond- 
ing vertical distances are k times smaller than those in I., while 
the velocities at corresponding points are the same. We will 
now prove that the system II. represents a possible state of 
motion, provided the density in II. is k times that in I. 

Let the function determining the state at a point P of the 
first gas be J?\(, ?/, f), and that at a corresponding point P' of 
the second gas J? 2 (, 7;, f). Then the condition for the density 
will imply that ,& ^ $.&,,,, Q. 

Since this holds for each group of molecules, the velocity of 
streaming, i.e. the mean value of f , will, as we assumed, be the 
same at P' as at P. 

We have now to ascertain whether the condition of a possible 
state of motion is satisfied, and for this purpose we consider the 
equation for a stationary state of the gas, viz. 

ap. dF IF Y 

ba=~ c + ^-17 +-*-- b 
dx dy ' dz 

If the Jf-axis is again drawn towards the right, and the F-axis 
perpendicularly to the plates, this equation becomes 



It holds for the state I. and, as is easily seen, also for the 
state II. In fact, for the state II. both sides of the equation 
become k 2 times greater. For the left-hand member expresses 



106 KINETICAL PROBLEMS CHAP. 

the number of collisions (and here the gas may be assumed 
homogeneous around P'), and if the number of molecules in each 
group is k times as large, the number of collisions is increased 
k 2 times. And with regard to the right-hand member, notice 
that rj is the same for both systems, and that dFddy^WFJdy, 
since F 2 =kF 1 and since all dimensions along the Y-axis in the 
system II. are k times smaller than those in the system I. Thus 
the equation of the stationary state within the gas is satisfied. 

It remains, however, to consider the state at the walls. The 
number of molecules of a given group which, per unit time, strike 
against a wall, will for the system I. be determined by ^F^dX, 
where F^ indicates the state at the considered point of the 
boundary layer. Similarly, this number for the system II. will 
be given by yFtflX. Suppose, further, that the two walls are 
perfectly equal ; then the collisions of molecules of the group 
considered will, in the system II., be k times more numerous 
than, but otherwise entirely the same as, those in the system I. 
Also for the molecules rebounding from the wall will the numbers, 
for groups with the same values of , rj, in the system II. be k 
times those in the system I. This, moreover, will be the case for 
perfectly reflecting as well as for diffusedly reflecting walls, the 
latter in contrast with the former being such that the molecules 
after reflection retain no trace of the mean motion relative to 
the wall which they had before the collision. If the above is 
assumed to hold also for the case that there is an adhering gas 
layer at the walls, then the system II. will represent also at tho 
walls a possible state of motion. 

Since the vertical dimensions in II. are k times smaller than 



those in the system I., we have (Fig. 2) B r D' = *BD or 

k 



Thus it is proved that the sliding is inversely proportional to 
the density. 

The friction, per unit area, upon a plane parallel to the 
-YZ-plane will, as we saw before (p. 101), be represented by 

w 
whence 



ii FRICTION AND SLIDING 107 

Since the velocity gradient in the system II. is k times that in I., 
and since w^p times this gradient, we have 



which proves that the friction is independent of the density. 



11. KUNDT AND WARBURG'S EXPERIMENTAL INVESTIGATIONS 

In 1875 an experimental investigation on friction and heat 
conduction in rarefied gases was published by Kundt and 
Warburg,* confirming the preceding theoretical results. 

A round horizontal disc S, in bifilar suspension, was made 
to oscillate in a gas between two fixed plates S l and S 2 . The 
coefficient of viscosity was determined by measuring the loga- 
rithmic decrement of the oscillations. This method was already 
applied by Coulomb in the case of a disc oscillating in an un- 
limited gas mass, while Maxwell improved it by introducing the 
two fixed plates, which has enabled him to make the calculation 
somewhat more accurate. 

Kundt and Warburg's formula is 



where M is the moment of inertia of the vibrating disc, D the dis- 
tance between the disc and one of the fixed plates, /8 the damping 
(appearing in the angular deviation w = ae~ ftt cos nt), and ft the 
viscosity coefficient, while is a number which depends on the 
density of the gas and the distance of the plates and which was 
accurately calculated by Maxwell. The spinning disc generates 
transversal friction waves. If the layer is thin enough, the air 
current is everywhere in phase with the disc, and = 0. Such was 
actually the ca^fe in Kundt and Warburg's experiments. Again, 
A would be equal to \irR* for each plate surface (of radius R) 
exposed to the friction of the gas, provided that the friction 
experienced by a surface-element of the disc having a velocity v 
is taken as equal to the friction upon a surface-element of 
an infinitely extended plate moving with the translation velocity 
v between two fixed unlimited plates placed at the same distance. 
The value of A was more accurately calculated by Maxwell, j" 

* Ann. Phys. und Chemie, civ., 1875, pp. 337 and 525. 
f Phil Trans., London, clvi., 1866, p. 249. 



108 KINETICAL PROBLEMS CHAP. 

In these experiments the pressure p of the gas and the 
logarithmic decrement X of the oscillations were measured. The 
latter decreases at smaller pressures, this being due to the 
influence of sliding. In fact, if for a large gas density, for which 
we can put i/=0, the logarithmic decrement is X , we have for X, 
which is proportional to the friction, for smaller densities, 

A D* /. 



since a 2v 

^ = jD' 

The factor a will, for a given distance of the plates, be a constant. 
Its value will vary inversely with this distance. 

Here are the results of some of the experiments in which the 
gas was air, and D =0-1104 cm. : 

P x 

320mm. 0-1318 

20 0-1300 

7-7 0-1292 

7-6 0-1292 

2-4 0-1256 

0-63 0-1109 

For a large number of observations on various gases and at 
different pressures, Kundt and Warburg actually succeeded in 
choosing a so that the values of X calculated by means of a, p, 
and X agreed with the observed ones. Some of these results 
were : 

p X obs. X calc. 

20 mm. 0-131 0-131 

7-6 0-129 0-129 

2-4 0-125 0-124 

1-53 0-120 0-120 

For different values of D the value of a appeared to be with 
fair accuracy inversely proportional to D. Thus for three 
different distances the values of a, when derived from the 
observations, were : 

0! =0-149, a 2 =0-070, a 3 =0-061. 
* cf . p. 105. 



ii FRICTION AND SLIDING 109 

On the other hand, starting from a and assuming that a a and 
a 3 are equal to a 1 D 1 /D 2 and a^D^D.^ the result was : 



a 2 = 0-084, a 3 = 0-059. 

For still smaller pressures we are reduced to the case in which 
the straight portion of the velocity-curve (Fig. 2) is absent. 
According to Kundt and Warburg, the formula for X continues 
to hold as long as the distance of the plates is at least 14 times 
the mean free path of the molecules. 

Lastly, they find for air at the pressure of 76 cm. and the 
temperature of 15 C. 

v = 0-00001 cm., 

which shows that the sliding at normal pressure is small. But 
at the pressure of 1 mm. one should then still find v = 0-0076 cm., 
so that for a distance D = 0-2 crn. of the plates the effect of sliding 
would already be very marked. Indeed, the term 2j/ in the 

expression ^ - could not be neglected in the presen.ce of D. 



The mean free path of the molecules at normal pressure is 
about 0-0000084 cm., and thus of the same order as v. 



CHAPTER III 
KNUDSEN'S INVESTIGATIONS ON RAREFIED GASES 

12. FLOW OF A RAREFIED GAS THROUGH A 
NARROW TUBE 

WHILE in what precedes we have considered the extreme case in 
which the gas density is large, we will now turn to Knudsen's 
investigations * which represent the other extreme case, corre- 
sponding, that is, to such a small density that the effect of 
collisions can be disregarded. Knudsen investigates, theoretic- 
ally as well as experimentally, the flow of such a strongly 
rarefied gas through a cylindrical tube whose walls are sup- 
posed to be so rough that the rebounding molecules have no 
trace left of their original streaming motion, being reflected from 
the walls in all possible directions. 

Obviously the problem in hand can be treated on similar lines 
to that of heat radiation. Analogously to the latter we will write 

A cos ddcoda 

for the number of particles rebounding from an element do- of the 
wall within a cone of aperture do>, the axis of the cone being 
inclined at an angle 6 to the surface normal. This amounts to 
assuming that, with respect to the reflected particles, the normal 
of the surface-element of the wall is privileged. The factor A 
depends on the total number of particles rebounding from the 
element da of the wall, and this is, for a stationary state, equal 
to the number of particles striking this wall-element. Thus A 
is a function of the state of the gas. 

* Ann. der Physik, xxviii., 1909, p. 75. 
110 



INVESTIGATIONS ON BAREFIED GASES 111 

Similarly as for heat radiation, the number of particles which 
two elements da- and da-' send to each other can be represented by 

A cos 0da>dcr = A cos O'dco'da' = A cos 6 cos 0'dcrdv', 

if& 

where r=PP' (Fig. 3). 

Thus, in a space occupied by a strongly rarefied gas a state of 
stationary equilibrium, analogous to that of black body radiation, 
is possible, provided the walls are kept 
at a constant temperature. For all ele- 
ments of the walls the value of A is the 
same, and the last expression 
holds also for the number of 
particles which pass through 
two surface - elements placed 
within the gas. 

We will assume for the gas * ^ FIG. 3. 

Maxwell's law of velocity distri- 

bution, and consider, in the first place, the number of particles 
which traverse, per unit time, a surface-element da- at a point P 
within the gas, and whose direction of motion falls within a cone 
of angle dw and axis coinciding with the normal of the surface- 
element. This number can be written 

Adada), 

and, if f(v)dv be the number of particles per unit volume whose 
velocity falls within the interval v to v + dv, 




00 

Ivd 
Jo 



j 

vdcr A - f(v)dv=Ad<jda>. 
o JX 

Thus, 



According to Maxwell's law, 



where h is inversely proportional to the temperature and G 
depends on the density of the gas. This gives 

G 



112 



KINETICAL PROBLEMS 



CHAP. 



Since the gas pressure is equal to two-thirds of the kinetic 
energy per unit volume, we have 



whence 



.JL A 

WWTty 7T 



Q 



Thus -4 is a function of the temperature [through h] and of the 
pressure. 

At constant pressure, A diminishes with increasing tempera- 
ture. This can be seen directly. In fact, at a higher temperature 
the molecules strike more vigorously ; if, therefore, the pressure 
is still to remain the same, there must be a smaller number of 
collisions. 

Let us now consider a gas flowing in a vertical cylindrical 
tube downward along the axis, as in Fig. 4, where the JK-axis 
is along the axis of the tube. 

We will assume the temperature to 
be throughout the same. If such were 
also the case of the pressure, the gas 
would remain in the state of equili- 
brium described above, and A would be 
throughout the same. But in our case 
the pressure increases up the tube, and 
the same will also be true of A. We 
suppose p to be slowly variable, so 
that it can be represented by a linear 
function of x, which amounts to re- 
taining of the Taylor-series development 
of p the first term only. In accordance 
with this assumption of small pressure 
variations, the values of A will be the 
same in all points of a cross-section of 

the tube. In fact, every surface-element can be assumed to be 
hit by as many molecules as if the pressure were everywhere 
the same as in the neighbourhood of that element. For the 
number of particles arriving from above is as much increased 
as that of particles moving upwards is diminished, so that the 




m INVESTIGATIONS ON RAREFIED GASES 113 

total number crossing the surface-element is not changed. A is 
thus a function of x alone, and this holds for every tube, no 
matter what the shape of its cross-section. 

Let us now consider an element da- of the section a? = placed 
at the point P (Fig. 4), and let us find the number of particles 
crossing this element per unit time in a bundle of directions 
contained within the cone of angular aperture da) whose axis 
makes an angle with the normal. This cone cuts the wall in 
an element placed at a point Q. Thus, the particles in question 
issue from an element d<r' of the wall within the cone aperture 
da', the axis of this cone making an angle 6' with the normal, 
so that their number can be written 

A Q cos 0'da)'dcr'. 

By what was said before this can be replaced by 

A v cos Odtodcr. 

Here A Q is the value of A at the point Q. Let Q Q be the 
intersection point of the generatrix of the cylinder [passing 
through Q] with the contemplated section, and let PQ = r. Then 

QQ = r cot0. 
Again, ^ 

A Q =A -r cot ' 9 
9 dx 

where A and dA/dx are the values of these magnitudes at the 
section x^=0. 

Thus the required number of molecules will be 

dA 

A cos Odwdcr - j- r cos 6 cot Odcodor. 
dx 

Integrating this expression with respect to da over a whole 
sphere, we shall find the current through the element da. Now, 
the first part contributes nothing to this integral, while the 
second gives 

- 2 -y- dajr cos 9 cot Oda), 

where the integral is to be extended over the upper half of the 
sphere. 

VOL. I I 



114 KINETICAL PROBLEMS CHAP. 

Thus we can write for the current across the whole section 



where 



cJA 

<* dx> 

cos 6 cot 6dw. 



Now, if <f> be the angle contained between r and a fixed direction 
in the section, 



dco = sin 



and 






cos 2 6d0 



(30) 



This integral is related to the self-potential of a substance 
uniformly distributed over the section. 
In fact, let p be the distance between 
two elements do- and dcr' placed at the 
points P and P' (Fig. 5). Then the 
potential at P will be 



FIG. 5. 




[ da ' --= r I T 
J p Jo h 



and if this be integrated over d<r, we have 



For a tube of circular section the integral is easily evaluated. 

Let (Fig. 6) be the centre of the 
circle and OA a fixed direction, so that 
ZJLP# = </>. Put OP = Z and OA=a. 
Then 



In (30) r can be replaced by 
provided we integrate over <j> from 
to TT. Thus we find 






//" 
da V 



_ 
- P sin 2 <f> d*f>. 




Now, taking for da- a ring of area SMI, we find 



in INVESTIGATIONS ON RAREFIED GASES 115 
Inverting the order of integrations we have 

fir/2 ra t 

(} = 47T 2 1 d<f> I \/fl 2 Z 2 sin 2 <f> Idl 
JQ Jo 

f 1 - cos tf> cos <f> sin 2 <f>] , . 

i " a ^i~"~ ~^" T~ i ' t d*r 

sin Q sin o I 

L ~ i J 



= * 3 7r 2 a 3 / f ---- -j- 
/0 2cos* 



+ cos < c&^ = j7T 2 a 3 tan + sin 



J 



7T/2 





Thus the total current across the circular section becomes 



-**% 



where 



?_ /* 

/ 

wir v^ 



If u* bo the mean squared velocity, then 

, 3 



and since the temperature is assumed to be throughout the same, 
we find for the current 

_8 /STT a 3 dp 
3"V 2 mu dx 

Here the current is expressed by the number of molecules. In 
order to express it in mass units, the last expression has to be 
multiplied by the mass m of a molecule. This gives 



_8 /37ra 3 d/) 
3 V 2 ti dx 



The volume of the gas flowing across the section per unit time, 
at the given temperature and pressure, will be equal to 'the last 
expression divided by the density />, i.e. 



3V 2 
This expression can be compared with that for the current which 



116 KINET1CAL PROBLEMS CHAP. 

would follow by applying Poiseuille's law to these small densities.* 
This would be, in terms of the volume, 

_7ra 4 dp 
Sp. dx 

where the friction coefficient /*, is approximately Q-3lpu\,1[ if p be 
the density, u the root of the mean squared velocity, and X the 
mean free path of the molucules. This gives for the current 
according to Poiseuille's law 

dp 



Apart from the numerical coefficients, the currents (31) and (32) 
are to each other as X to a, that is to say, as the mean free path 
of the molecules to the radius of the circular section of the tube. 

Since in our case X is very large compared with a, the 
amount of gas flowing through the section per unit time will be 
much greater than according to Poiseuille's law. In fact, it 
appears from Knudsen's experiments that it may well be 50,000 
times as large. 

Knudsen gives the current in terms of volume, at a pressure 
of 1 dyne per cm. 2 and a temperature r of the gas. If p l be 
the density of the gas in these conditions, the current is expressed 

by 

_8 /37T a 3 dp 
3 V 2 Piw dx 



The relation p = $pu z gives in the present case J-p 1 u 2 = l or 
u = VS/ft. Thus, if PQ be the density at the pressure 1 and the 



temperature 0, and therefore pi = , ---- (a = 1/273), we have 

ultimately the current 

,,dp 
~ C dx' 
where 



r = ? /TT cP^/l+ar 
3V 2 v " 



* Cf. p. 81, equation (7). 
t Cf. Maxwell, Phil. Mag. (4), vol. xix., 1860, p. 31. 



in INVESTIGATIONS ON RAEEFIED GASES 117 



13. KNUDSEN'S EXPERIMENTAL INVESTIGATION ON THE FLOW 
OP A RAREFIED GAS THROUGH A NARROW TUBE 

In Knudsen's investigation this formula was tested experi- 
mentally. In his experiments the stream passed through a tube 
joining two vessels filled with a rarefied gas. Each vessel was 
connected with a MacLeod manometer by means of which it was 
possible to measure very small pressures by compressing the gas 
to a known small volume, measuring its pressure and thence 
calculating the original pressure by Boyle's law. For pressures 
smaller than 5 mm. the pressure in each vessel is in this way 
measured separately, while for greater pressures the pressure 
difference between the two vessels is measured directly. 

Let the volumes of the vessels be v l9 v 2 and the pressures 
within them p and p& and let p^ > p 2 . For a cylindrical tube of 
length I 

dp_]^--p 1 
dx~ I ' 
whence the current 



. 
dx I 

The volume of gas flowing through a cross-section of the tube 
per second is, when measured at the pressure 1, 



these being the quantities leaving, per second, the first and 
streaming into the second vessel, so that 



Whence 



or 



log (P! -p z ) - - -j-(- +-) t+ const., 



Pi ~ Pa 



118 KINETICAL PROBLEMS CHAP. 

Since 



^2 

*T " A 



the constant C can be determined from the change of the pressure 
difference during a given time. 

Knudsen determined from his observations the quotient 



whose theoretical value is 
8 



2~~ (33) 



In his experiments the value of A/a mounted up to 6000. He 
worked with hydrogen, oxygen, and carbon dioxide. 

In order to find the effect of the length of the tube, different 
tubes were used, of which the first was about 6 cm. long and 
about 0-02 cm. in diameter, while the second was about twice as 
long and had the same diameter. The gas in this case was 
hydrogen. 

The ratio of the observed values of C" thus found was 1-95, 
while according to formula (33) it should be 2-05. 

Next, to find the effect of the cross-section, carbon dioxide was 
passed through the first and through a third tube whose length 
was about twice and diameter about 1 -4 times that of the first 
tube. The ratio of the experimental values of C' was 1-20, 
while th$ formula gave 1-15. Whence it appears that the 
current is actually inversely proportional to Z and directly pro- 
portional to a 8 . 

In order to investigate the effect of the density of the gas, a 
set of 24 parallel tubes were used, each 2 cm. long and about 
0-006 cm. in diameter. 

According to the formula C' should be inversely proportional 
to Vp Q and, therefore, C'Vp* should be independent of the kind 
of the gas. That such is actually the case, will be seen from the 
following results : 

Gas C'(obs.) V po CVpo 

Hydrogen . . . 0-168 1 0-168 

Oxygen . . . 0-0409 4 ' 0-164 

Carbon dioxide . . 0-0348 4-69 0-163 



in INVESTIGATIONS ON RAREFIED GASES 119 

To find the influence of the temperature of the gas, Knudsen 
made use of the first tube and filled the vessels with hydrogen 
at a mean pressure of 0-3 mm. In the results of the experiments 
the gas volume flowing through the tube is reduced to the 
temperature 0, so that 

8 A 3 

3V 
Consequently, 

G'Vl 



should, according to the theory, be independent of the tempera- 
ture. Now, Knudsen found : 

Temperature G' C\ / - 1 ar - 

1 Vl+22a 

22 0-0713 0-0713 

100 0-0011 0-0721 

196 0-0588 0-0741 

This table shows that also the dependence of C' on the tempera- 
ture is correctly represented by the formula. 

Finally Knudsen proved the correctness of the constant factor 
in the formula by actually determining in different cases the 
absolute amount of the gas streaming through and by comparing 
the results with those calculated by the formula in question. 
His results were 

Tubes Gas C' (obs.) C' (ealc.) 

1 Hydrogen . . 0-073 0-080' 

1 Oxygen . . . 0-0187 0-0202 

1 Carbon dioxide . 0-0166 0-0172 

2 Hydrogen . . 0-0375 0-0392 

3 Carbon dioxide . 0-0199 0-0198 

4 Hydrogen . . 0-168 0-161 
4 Oxygen . . . 0-0409 0-0404 
4 Carbon dioxide . 0-0348 0-0344 

With reference to this very rarefied state of the- gas Knudsen 
speaks of a " pure molecular streaming." He extends also his 
researches to the case of larger densities for which the collisions 
of the molecules can no longer be disregarded and which offers 
considerable theoretical difficulties. In the latter case Knudsen 
speaks of " mixed molecular streaming." 



120 KINETICAL PROBLEMS CHAP. 

14. FLOW OF A RAREFIED GAS THROUGH A NARROW ORIFICE ; 
KNUDSEN'S EXPERIMENTS 

A second investigation due to Knudsen * concerns the flow 
of a gas through a narrow orifice. 

We assume again that the gas is very strongly rarefied, so 
that the mutual collisions of the molecules can be disregarded, 
and that the dimensions of the orifice are small compared 
with the free path of the molecules. We suppose, further, that 
the temperature of the gas is throughout the same and that the 
diaphragm with the said orifice divides the space occupied by the 
gas in two portions, so that the pressure above the diaphragm is 
throughout p l9 and below the diaphragm p 2 . 

By what was said before the number of particles passing, 
per unit time, through an element da- of the orifice from the 
upper to the lower chamber in a direction contained within a 
cone of angular aperture da) can be written 

A 1 cos 9 d<x)d<r, 

where is the angle between the axis of the cone and the normal 
of the surface element, and A t a factor depending on the tem- 
perature of the walla of the upper chamber and on the pressure 
of the gas therein contained, but independent of the cone chosen. 
Thus the number of particles flying downwards through d<r per 
unit time will be 



r/2 
^irA^ cos 9 sin 9d9da = 7rA 1 dcr, 
. 

and therefore the total number through the whole orifice i, 
Similarly the number of particles flying, per unit time, upwards is 



Whence, the resultant flux downwards, measured by the number 
of molecules, - 


Since 



- 

7T 2U 2 



* Ann. der Phys., xxviii., 1909, p 999. 



in INVESTIGATIONS ON RAREFIED GASES 121 
the last expression becomes 



As before, Kundsen expresses the flux through the orifice per unit 
time in terms of the volume the gas would occupy at a tempera- 
ture r and a pressure of one dyne per cm. 2 . If p l be in this case 
the density, then w = \/3//^ 1 (cf. p. 116), and the expression for 
the flux becomes 



/ i 



This formula was again tested experimentally by Knudsen who 
used the same apparatus as in the preceding investigations, but 
having replaced the original tube by a glass tube containing a 
plate of platinum in which an orifice was pierced by means of 
a fine needle. He worked with orifices of an area of 0-0005 mm. 2 . 
According to the last formula the flux is proportional to p -p 2 
and can thus be written 



the theoretical value of G being %lV%irpi. The experimental 
determination of C proceeds again on the same lines as in the 
previous experiments. In fact, if t? L and v 2 be the volumes of 
the gas at the two sides of the orifice, the volume of the gas 
streaming through per unit time, referred to the pressure of 1 dyne 



per cm. 2 , is 



whence 



The following are some of Knudsen's results 



Gas 

Hydrogen 
Oxygen . 
Carbon dioxide 


C (calculated from 
observations) 

. . 0-225 
. . 0-0565 
. 0-0465 


C (by theoretical 
formula) 

0-236 
0-0576 
0-0491 



These results give, in Knudsen's words, an experimental 
confirmation of the correctness of the kinetical theory of gases, 



122 KINETICAL PROBLEMS CHAP. 

and especially of the validity of Maxwell's law of the distribution 
of velocities. If one assumed against Maxwell's theory that all 
molecules have the same velocity, one would find from the 
theoretical formula for the coefficient C values by 8-6 per cent, 
greater than those of the above table. The observed values of C 
would then differ from the theoretical ones by amounts which 
could not be thrown upon the experimental errors. 

Also in the case of a gas flowing through a narrow orifice 
the phenomena are different at a higher pressure and their theory 
becomes difficult. We have then to distinguish between the 
case in which the mean free path is comparable with the dimen- 
sions of the orifice and that in which it is small compared with 
the latter, which finally leads to the formation of jets. 



15. FLOW OF A RAREFIED GAS THROUGH A NARROW TUBE 
WHOSE ENDS HAVE DIFFERENT TEMPERATURES 

We will now return to the Jlwo of a gas through a tube 
in order to investigate the effect of temperature differences at 
different places along the tube. 

By what was said before (p. 115), the current is determined by 

8 2 3 A A 

rt 7T 2 tt 3 j 9 

3 dx 

where 

A 

** = 

and it can therefore be written 

(p 



_8o 3 /3 
3 m V " 



3 m V "2 dx 

The current is thus dependent on the pressure gradient and the 
temperature gradient. Consequently, the condition for the 
absence of streaming in the tube is 

^ = const., 

u ' 

or p 

--= const. 



Thus we see that, while in a wide tube, in comparison with 



m INVESTIGATIONS ON RAREFIED GASES 123 

whose dimensions the mean free path of the molecules is small, 
the condition of equilibrium is that the pressure should be every- 
where the same and, therefore, the density inversely proportional 

with the absolute temperature T (since ? v = const.), in the 

present case the pressure must be proportional to VT and, there- 
fore, the density inversely proportional to VT. 

In a narrow tube whose extremities have different tempera- 
tures these pressure and density differences will arise by them- 
selves. In fact, if originally the pressure is everywhere the same, 
the gas will stream, according to the above formula, towards 
places of highest temperature, so that the pressure will increase 
there. 

This was verified by Knudsen * experimentally. He con- 
nected two MacLeod manometers by a set of tubes of different 
cross-sections. In his first experiments a single capillary tube 
only was used, 9 cm. long andO-6 mm. in diameter. This capillary 
was placed between two tubes of 14 mrn. in diameter. The 
junction of one of the wide tubes with the capillary was brought 
to a temperature of about 350, which raised the other end of 
the capillary to a temperature of about 100. The determination 
of these temperatures, however, was uncertain. The two mano- 
meters could also be connected with each other directly by a 
wide tube. The purpose of the observations with a direct 
connection, when the pressure in the two manometers should be 
found equal, was only to form an idea of the accuracy of the 
measurements. The results obtained for hydrogen were : 

Pi PI 

For direct connection . . . 0-0218 0-0216 
For connection by capillary tube . 0-0223 0-0211 

There is thus, in fact, a pressure difference between the two 
ends of the capillary tube. The whole system shows much 
similarity to a thermo-element. If the two wide tubes are, 
besides the capillary, connected also by a wide tube, the gas will 
continue to flow, and in such a direction that the stream in the 
capillary is from a lower towards a higher temperature. This 
stream is thus analogous to a thermo-electric current. The effect 

* Ann. der Phys., xxxi., 1910, p. 205. 



124 KINETICAL PROBLEMS CHAP. 

will also here be increased by connecting the manometers by a 
large number of wide and of narrow tubes, alternately arranged 
(Fig. 7). At the places marked in the figure by rectangles the 
tubes were heated by means of platinum wires which were brought 
to glowing by an electric current. The temperature difference 
between the heated and the not-heated contact places of the 
tubes was measured by means of a thermo-element and amounted 
to about 500. The following wore the results for hydrogen : 

Pi . , lv Pi (calculated 

Pi V* --(observed) ^ theoretically) 

0-00978 0-00419 2-33 2-49 

The pressure difference for different temperatures in different 
parts of a tube filled with a strongly rarefied gas was demon- 
strated by Knudsen in yet another way.* In a glass tube of 



Tb- 



Km. 7. 



7-5 mm. in diameter magnesium powder is placed between 
asbestum stoppers. One end of the magnesium column is heated 
electrically up to a temperature of 248^ which gives to the other 
end a temperature of 22. The pressures at the two ends of 
the tube are measured by means of MacLeod manometers. The 
equilibrium in this case sets in very slowly. Finally we should 
have R _ I T \ 

ft~v*V 

where p t and p 2 are the measured pressures. 

Knudsen found for p and p 2 , after different times : 

Time Pi Pz 

0-3000 0-3000 

18* 25' 0-3308 0-2716 

23* 25' 0-3347 0-2693 

46 h 50' 0-3414 0-2641 

oo 0-3428 0-2628 

The last values of p l and p 2 which would correspond to a perfect 

* Ann. der Phya., xxxi., 1910, p. 633. 



in INVESTIGATIONS ON RAREFIED GASES 125 

equilibrium were obtained by extrapolation, it being assumed 
that the relation 



is valid, q being the pressure difference in the state of equilibrium 
From what precedes it is clear that a porous plate should 
show the following effect. If one face of the plate is warmer 
than the other, air should stream from the colder to the warmer 
face, and this flow should continue so long as there is a tem- 
perature difference between the two faces. 

This phenomenon was demonstrated by Knudsen * in a simple 
way. In a vessel of porous material air is heated electrically. 
Thus the inner surface of the vessel is brought to a higher 
temperature than its outer surface, and air is being drawn 
through the walls into the vessel. If, therefore, the vessel be 
fitted with a tube ending in a bottle containing water, air will 
be seen to bubble up through the water. Knudsen found that 
through a vessel of the capacity of 100 cm. 3 as many cubic 
centimetres of air could easily be drawn. If the air cannot 
escape, a pressure difference is produced. With a strong heating 
the pressure within the vessel is a few centimetres (of mercury) 
higher than outside. 

Knudsen points out that such phenomena should happen 
often, and that also in the case of greater gas densities a 
temperature difference will bring about a pressure difference. 
According to Knudsen this phenomenon is undoubtedly of great 
importance in nature. It plays perhaps a role in the respiration 
of plants and contributes to the refreshing of air in the "porous 
soil, air being given out where the surface of the earth is heated 
by the sun, and sucked in where no such heating takes place. 

16. MUTUAL REPULSION OF Two PLATES AT DIFFERENT 
TEMPERATURES SEPARATED BY A RAREFIED GAS 

Knudsen f has constructed an absolute manometer based upon 
radiometric action. This was suggested to him by an investiga- 
tion on the change of the velocity of gas molecules due to their 
impact against a fixed wall differing in temperature from the 

* Ann. der Phys., xxxi., 1910, p. 207. 
t Ibid., xxxii., 1910, p. 809. 



126 KINETICAL PROBLEMS CHAP. 

gas. This occurs already in Crookes' radiometer. In this 
apparatus four vanes of mica, covered on one side with lampblack, 
are mounted on the arms of a horizontal cross free to spin round 
on a pivot. If the mutual collisions of the air molecules are 
disregarded, the following explanation of the motion of the 
radiometer seems to be the simplest. Both faces of each vane 
are hit by equally numerous molecules and in like manner, but 
those molecules which impinge against the blackened faces will 
be thrown back with a greater velocity than those hitting the 
clear faces (for, owing to a greater absorption, the former have 
a higher temperature). As a result the vanes will revolve with 
their unblackened faces turned forward. To many physicists, 
however, it seemed doubtful whether this simple theory could 
be here applied. 

Now, Knudsen considers a case for which the theory can be 
adequately developed. He takes two plates S and S' (Fig. 8) 
immersed in a strongly rarefied gas and placed at a distance 
from each other which is small in comparison with the mean 
free path of the molecules. The plate S has the temperature of 
the surrounding medium, while S' is brought to a higher tempera- 
ture, so that the plates will exert upon each other a repelling 
force. This force is a function of the pressure. The latter can 
therefore be determined by measuring the former, which gives 
the principle for constructing a manometer. 

We saw from the preceding discussion of the flow of a 
gas through narrow tubes and small orifices that a temperature 
difference gives rise to a pressure difference. Let p be the 
pressure between the plates, p the pressure outside, and T and 
T' the temperatures of the two plates. Then Knudsen takes for 
the temperature between the plates the mean %(T + T') and 
derives the relation __ 

P. ^v%( T + T ) 
Po VT 

If K be the repelling force per unit area, this relation gives 




jr 

and p -- j=- ..... (34) 



IT+T' ' 
V 2f~~ 1 



m INVESTIGATIONS ON RAREFIED GASES 127 

The formula used by Knudsen in his experiments with the 
manometer differed from this somewhat ; it was 



The force K is measured by means of a torsion balance to 
which the plate S is attached, while the fixed plate S' is heated 
electrically. 

Knudsen subjects the formula (34') to a further scrutiny, 
but we shall not follow him in- this and shall consider the question 
in a somewhat different way. 

We assume that the dimensions of the plates are large com- 
pared with their distance, that the molecules move from one 
plate to the other in all directions according to 
the cos 0-rule (cf. p. 110), and that Maxwell's 



velocity distribution holds. 



T' 



Let us now follow a molecule on its path 
through a unit time, during which interval it 
moves many times back and forth, and as many 
times towards the right as towards the left. We 
will calculate the number of flights in a certain FI. 8. 

direction, say, towards the right, which we will 
denote by v. The number of flights for which the velocity of 
the molecule is contained between v and v+dv and the angle 
of inclination of this velocity to the normal erected upon the 
plates between and 0+d0, will be found by multiplying the 
number v by the probability that v and should fall within 
these limits, that is to say, by the fraction of all molecules 
rebounding, per unit time, from one of the plates, for which the 
velocity and direction of motion satisfy these conditions. 

The number of molecules, per unit volume, whose velocities 
lie. between v and v + dv is proportional to e~ hc *v z dv, and the 
number of all such molecules leaving per unit time one of the 
plates is equal to Ce~*"iMv, 

where h corresponds to the temperature of that plate. The 
constant C must satisfy the condition 



whence 



of. 

Jo 



128 KINETICAL PROBLEMS CHAP. 

The requirement has still to be satisfied that the direction of 
motion should make with the plate normal an angle included 
between 6 and 0+d0. The number of particles leaving the plate 
is, with equal cone aperture, assumed to be proportional to 
cos 0. Again, the (angular) space between two cones of apertures 
6 and 9+d0 is proportional to sin Odd. Thus, the probability 
that the molecules will have the required direction of motion, 

C' sin 6 cos i 



where C' is determined by the condition 

r/2 
Bin0cos0cZ0 = l, 

or C"=2. 

Since these two probabilities are independent of each other, 
the required probability will be equal to their product, i.e. to 

l 



4 A 2 sin cos 6e 

Multiplying this by v we shall find the number of passages 
towards the right made by a molecule per unit time under the 
said conditions. The same expression will be found for the 
corresponding number of passages towards the left, provided h is 
replaced by h', where h' belongs to the temperature T r of the 
heated plate. 

An expression for the number v itself will now be found by 
noticing that the time taken by a single passage is lj(v cos 0), 
where I is the mutual distance of the plates, and that the time 
taken by all the passages under consideration is the unit of time. 
Thus, 

H- 

Since f* ~A<' 1 

/ c v*dv = 
JQ 4A 

this gives 

= -- - 



\lv sin OMpeiMv+ sin 0de h'*e~~ h ' v 'v*dv 



In order to find the pressure between the plates we have only 
to take the difference of the components along the plate normal 
of the momentum of all the molecules flying away, per unit of 



in INVESTIGATIONS ON RAREFIED GASES 129 

time, from the plate S and of all those flying towards the plate S. 
These two components have opposite signs. Consequently, the 
pressure will be given by the sum of their absolute values. 

For the number of times a molecule, within the given limits 
of v and 0, rebounds from the plate S, we have found, per unit 
of time, 

4vh* sin cos Oe'^'iPdOdv. . . . (35) 

With this direction of motion and this velocity the component of 
momentum along the plate normal is mv cos 6. Thus, multiplying 
the expression (35) by mv cos and integrating over all values 
of v and 0, we shall find the momentum carried by a particle 
which rebounds from S, per unit time. Multiplying this further 
by the total number of particles, we shall have the pressure 
exerted upon the plate by the molecules which move towards 
the right. For the pressure due to the molecules moving towards 
the left a similar expression holds, with Ji replaced by h'. Divid- 
ing by the area of the plate, we shall have the pressure per unit 
area. Thus, if n be the number of molecules per unit volume 
and Z the distance of the plates, the pressure will be 

p = 4vmnl\ r ~ sin 9 cos 2 Od8 j We~ hv \\lv 



sin cos 2 

o 



mn 



Since 

we have ultimately 



3 
In order to verify this result we can put h-h r . Since A = 9 ~ 

wehave mn 1 



which is the correct value of the pressure when both plates have 
the same temperature. 

Thus the pressure in the space between the plates is deter- 
mined. The pressure p Q outside the plates is smaller, and the 
difference p -p gives the repelling force K. 

K 



130 KINETICAL PROBLEMS CHAP. 

The temperature is throughout the greater part of the space 
outside the plates equal T, and the density is only in a thin 
layer adjacent to the plate S' smaller than in the remaining 
space. If this thin layer be disregarded, the number of molecules 
per unit volume can be put equal w and the pressure 



Thus we find 

mn 

(36) 

It remains to establish a relation between n and w . For this 
purpose suppose first both the plates had the same temperature T. 
Then the density of the gas would be everywhere the same. If 
now the plate S f is heated up to the temperature T', the first 
consequence will be that at the edge of the plates more particles 
will pass from the space between the plates into the outer space 
than vice versa, until an equilibrium is reached and as many 
particles pass in one as in the other direction. A relation 
between n and n Q will thus be found by equating to each other 
the numbers of particles moving in one and in the opposite 
direction. Since, however, the state of motion near the edges is 
complicated, we will simplify the reasoning by an artifice, viz. 
by supposing that there is an orifice in the plate S which is so 
small as not to change the state perceptibly. Then, in the case 
of equilibrium, also the number of molecules passing through the 
orifice into the space between the plates will be equal to the 
number of those passing through the orifice outwards. Let a) 
be the area of the orifice. In the first place, the number of 
particles passing through the orifice outwards, per unit time, is 
equal to the number of those which per unit time strike the area 
ft> of the plate S. By what precedes, this number is 

wn 



Next, to find the number of particles passing per unit time 
inside, notice that the orifice is made in the non-heated plate, so 
that in the space outside the plates and also at the surface of the 
plate S the temperature can be taken to be T and the number 
of particles per unit volume w . The number of particles per 



m INVESTIGATIONS ON RAREFIED GASES 131 

unit volume having a velocity between v and v+dv and a 
direction of motion between 6 and +d0 is Cn sin tie~ ho *v z d0dv, 
where C is determined by the condition 

fir r 00 

Cn Q l amOdOl e 
JQ Jo 

which gives 



0=2 



V w' 



The number of particles per unit volume satisfying the said 
condition for v and is, therefore, 



7T 



. (37) 



The number of particles striking, per unit time, the area o> 
will be the product of (37) into w cos integrated over all v and 
over from to \tr, i.e. 



2. l h 'n^r sin cos ftfll Y%Vi,- ?"-. 
V 7T A) .'o 2v^A 

Equating this number to that found above for the number of 
particles passing through the orifice towards the left, we find 



or n 



Substituting this into (36), we have 



or, since A/A' =T'/T, 

- 1 ..... (38) 



which is the formula used by Knudsen (cf. (34'), p. 127). 

ItT'-T is small, this expression, as well as (34), reduces to 

T' -T 

K= p-" ..... < 39) 



132 KINETICAL PROBLEMS CHAP. 

Notice that in deducing (38) it was assumed that every 
particle after striking a fixed wall acquires a velocity which 
corresponds to the temperature of the wall. From Knudsen's 
later investigations it appears that this is not the case, and that 
the mean variation of the kinetic energy is only a fraction of 
what it would be according to our assumption. 



17. KNUDSEN'S MANOMETERS 
Upon the formula - 



is based the use of Knudsen's manometer. Here p Q is the total 
pressure, including that of the mercury vapour. The formula 
holds for P Q <^QQ mm. of mercury (4 to 5 dynes per cm. 2 ). For 
greater pressures we can take 

2K 



~ If' 

\ f~ 



where c> 1. The dependence of p Q upon the temperature 
remains as in the original formula, since c is independent of 
temperature. 

The first apparatus used by Knudsen consisted of a platinum 
strip S' which was fixed and could be heated electrically, 
the temperature being determined by measuring the resistance. 
At a short distance from this was placed a platinum plate 
suspended on the arm of a torsion balance by means of which 
the force K could be measured. The whole apparatus was 
placed under a jar in communication with a Gaede mercury pump. 

The following are the results of some experiments in which 
the plates were placed in air : 

T ..... 31-7 118-9 198-5 274-5 376-2 

T' ..... 23-4 23-7 24-7 28-0 37-3 
p in dynes per cm. 2 (cal- 

culated from K) . . 2-31 2-45 2-27 2-21 2-31 

Since the pressure between the plates was kept constant as far 
as possible, these numbers give a good verification of the formula. 



m INVESTIGATIONS ON RAREFIED GASES 133 

The mean of the ;p -values is 2-28 dynes per cm. 2 , while the 
measurement with a MacLeod manometer gives 0'20 dyne per 
cm. 2 . To the latter must be added the pressure of the saturated 
mercury vapour, for which Knudsen's previous measurements,* at 
a temperature of about 23, gave 2-04 dynes per cm. 2 . Thus the 
total pressure becomes 2-24 dynes per cm. 2 , which agrees well 
with the pressure found with the new apparatus. Knudsen has 
repeated these measurements with hydrogen. 

The condition that the plate distance should be small 
compared with the mean free path X of the molecules was 
satisfied. In fact, since at a pressure of 1 dyne per cm. 2 the 
mean free path X amounts to about 10 cm., in the experiments 
under discussion X was about 4 cm., while the distance of the 
plates amounted only to 0-055 cm. 

Other forms of manometers are described in the quoted paper 
by Knudsen. One of these apparatus consisted of a copper 
cylinder with a polished end-surface which served as the heated 
plate. The cylinder is heated by an electric current, and the 
temperature is measured by means of a mercury thermometer 
placed within a cavity of the cylinder. Opposite the polished 
end-surface of the cylinder there is a copper plate attached to 
the bulb of a thermometer which is suspended on the 
arm of a torsion balance. The copper cylinder is 
surrounded by sheets of copper which are not heated, 
and serve as guard rings. By means of a special 
system of pipettes Knudsen was able to introduce into 
the testing space exceedingly small quantities of gas. 
For measurements at high temperatures he used an 
apparatus of platinum. 

Even at very high temperatures the formula (38) 
turned out to agree fairly well with the experimental 
results, while this did not seem to be the case with the 
approximate formula (39). 

Yet another very simple apparatus consisted of a ,7" ' 
wide glass tube A (Fig. 9), into which a second glass F IG . 9. 
tube 6 was sealed. The tube 6 had on one side, at c, 
an aperture. Opposite this, in the middle of the tube, was 
suspended on two cocoon threads a plate of mica, P. When the 
apparatus is placed in warm water, the plate of mica experiences 
* Ann. der Phya., xxix., 1909, p. 179. 




\ 



134 KINETICAL PROBLEMS CHAP. 

a repulsion, due to the difference of its own temperature and of 
that of the outer wall. From the positions assumed by this plate, 
when the apparatus was placed in cold and then in warm water, 
the repulsion, and thence also the pressure of the gas, could be 
measured. After a certain time, when also the innermost tube 
reached the temperature of the bath, the repulsion ceased.* 

18. KNUDSEN'S ACCOMMODATION COEFFICIENT 

In a further paper, entitled " The Molecular Heat Conduction 
of Gases and the Accommodation Coefficient,"! Knudsen points 
out that the velocities with which the molecules, impinging upon 
a fixed wall, rebound from it do not correspond to the tempera- 
ture of the wall. He introduces, therefore, an accommodation 
coefficient a, by which the temperature variation which the 
molecules would undergo at the impact, if they assumed the wall 
temperature, has to be multiplied in order to give the true 
temperature variation. The accommodation coefficient depends 
on the nature of the gas and of the wall. The more rough the 
wall, the greater the value of a, tending to 1. 

Let us once more consider the two plates S and S' (Fig. 8) 
at the temperatures T and T'. Then the molecftiles which fly 
away from S towards the right will not have, as previously 
assumed, the temperature T but some other temperature 0, and 
similarly those moving away from S' towards the left will have a 
temperature 0' instead of T' 9 where 6 and 6' lie between T and T 1 . 

By the definition of the accommodation coefficient we have 
the relations 6' =a(0' T} 

0-e i =a(B-T'), 

whence = T + ~- a (T' - T), 

2-a v ' 



* For the sake of completeness it may be mentioned that Langmuir (Phy*. 
Rev. (2), 1, 1913, p. 337) constructed a manometer based on the properties of 
rarefied gases, by means of which pressures as small as 10~ 7 mm. of mercury 
could be measured. 

f Ann. der /%., xxxiv., 1911, p. 593. 



m INVESTIGATIONS ON RAREFIED GASES 135 

The accommodation coefficient is here assumed equal for the 
two plates. In calculating the pressure the velocities of the 
molecules can be taken as if a 1, provided that T and T are 
replaced by 6 and 0'. 

Whence it follows that in the expressions 

mn 



A corresponds to the temperature T, while h and h' correspond 
to the temperatures 9 and & respectively. 

In order to determine the force K we again assume the 
presence of a small orifice in the plate 8 and introduce the 
condition that, per unit time, as many molecules should pass 
through w into the space between the plates as cross it outwards. 
This gives the equation 



Thus, 



and 



The circumstance that the accommodation coefficient is not equal 

to unity affects only the term ---_+ . Since h is inversely 

Vh ' 



proportional to the absolute temperature, we can put 

7 C j t 

*>*-? 

Now, T + T' = 6 + 0'. Consequently, in the expression 



and 0' can be replaced by T and T', provided the temperature 
difference is small. Only for greater temperature differences must 
the accommodation coefficient be taken into account. 



136 KINETICAL PROBLEMS CHAP. 

19. HEAT CONDUCTION IN A RAREFIED GAS CONTAINED 
BETWEEN TWO PLATES OP UNEQUAL TEMPERATURE 

We now pass to Knudsen's investigation on heat conduction * 
and, for this purpose, consider again the plates S and S' (Fig. 8). 
We have to calculate how much energy (for polyatomic gases 
including also the internal energy of the molecules) is transferred, 
per second, from one plate to the other. We know from previous 
considerations that a molecule flies away from the plate S a 
number v of times per second, where 

( Cf -P- 128 -) 

The probability that the velocity of such a molecule is con- 
tained between v and v + dv and that the direction of its motion 
makes with the plate normal an angle falling 'between and 
+ d0 was already deduced and amounts to 

47* 2 sin 6 cos 0e~ hl \*d0dv (cf. p. 129), 

where h is determined by the temperature of the plate from which 
the molecule rebounds. 

We now introduce the new condition that the internal energy of 
the molecule should lie between e and e +rfe. Let the probability 
of this be represented by f(h, e) de, where h indicates that the 
function / depends on the temperature. Since, by Boltzmann's 
distribution law, the function/is independent of v, the probability 
for all the three conditions to be satisfied is equal to the product 
of the probabilities of each of them, so that the number of times 
a molecule rebounds from S under the said conditions is 



4v sin cos 0A 2 T A 'V/(A, )d0dvde. 
Here the function /(A, e) satisfies the condition 



If h is the mean internal energy at a temperature corresponding 

tO A, 



* Ann. der Phya., xxxiv., 1911, p. 593. 



in INVESTIGATIONS ON RAREFIED GASES 137 

Since a molecule leaving the plate carries with it the energy 
e, we find for the total energy transfer, per unit of time, 



v f 
JQ 



/2 sin cos 0d9 C fW Al V(Jwt> 2 + )f(h, )dedv. 
Q Jo -o 



The integration over 6 gives 



/o -o 
and that over v, 



snce 



f .-'Vtf.-i f -""4 
Finally, an integration over e gives 



If w is the number of molecules per unit volume and / the distance 
of the plates, the flux of energy per unit area will be 



m - 



The energy gained by the plate, per unit area and unit time, 
from the molecules moving in the opposite direction is given by 
the same expression, only with h replaced by h' in the last factor. 
Ultimately, therefore, we find for the heat transfer from the 
plate S' towards the plate S 



n 



Since u 2 = 3/2/j, the term m/h can be replaced by e oh , where e oh 
is the mean kinetic energy of translatory motion at a temperature 
corresponding to h. Similarly m/h' will be replaced by J oh >. 
The heat transfer from S' to S can thus be written 

n /4 4 _ 



138 KINETICAL PROBLEMS CHAP. 

The factor shows that in the process of heat transfer the 
energy of translatory motion plays a greater role than the internal 
energy, which can easily be explained, since with a quicker 
translation the molecules move also more often to and fro between 
the plates. If the temperature difference T - T is small, the 
last expression can be written 



This holds also for greater temperature differences provided 6 
and e are linear functions of T, which is the case of e . 

The quantities d JdT and d^dT are related to the specific 
heats. If c v be the specific heat at constant volume, and c p that 
at constant pressure, both per unit mass, then 



whence 

_ 

~m\3~dT 



Ultimately, therefore, the expression (40) becomes 



which is proportional to the number of molecules per unit volume, 
and thus to the density. 

Knudsen writes for this heat transfer per unit area 

tf-<r-r)3*, f .... (42) 

where p is the pressure (there being, for a small temperature 
difference, no need for distinguishing between the pressures at 
T and T') and e x is a coefficient introduced by Knudsen which, 
by (41), has the value 



in INVESTIGATIONS ON RAREFIED GASES 139 



Since h =pj2p and mn =p, while Vh + Vfr can be replaced by 
, this expression reduces to 



If M be the molecular weight of the gas and R the gas constant 
per gram molecule, then 

RT 

*-ir* 

c -c = R - 

and therefore, 

R 3 c +c 



The heat transfer was thus far expressed in mechanical units. 
To express it in calories, we have to take for e K the last value 
divided by E 9 the mechanical equivalent of a heat unit. With 
E = 419 . 10 5 and R - 83-2 . 10* we find 



With a somewhat different deduction of the formula Knudsen 
finds for the numerical factor the value 4346. 10~ 6 . 

We have thus found for the number of calories transferred 
per unit time and unit area from the plate S' to the plate S 



The heat transfer appears then to be independent of the 
distance of the plates. 

This formula can also be used to represent the heat transfer 
in other cases, provided the temperature difference T r - T is 
small. Thus, for instance, we can imagine a hot body A (Fig. 10) 
kept at the temperature T' and surrounded by an enclosure B 
having a lower temperature 7. Since all molecules impinging 
against A arrive from B (mutual collisions of molecules are 
excluded), they will have the energy corresponding to the 
temperature T, while the molecules coming from A have the 
energy e' corresponding to T', the accommodation coefficient 




140 KINETICAL PROBLEMS CHAP. 

being for the present left out of account. The heat supplied by 
A can thus be represented by 



if N be the number of molecules received or sent out by A per 

unit area and unit time. The temperature difference being 
small, the distinction between the .AT- values 
corresponding to the temperatures T and 
T' is of no account, and we can take the 
number which would hold for T' = T. This 
number will be proportional to the pressure 
and independent of the shape and the 
dimensions of A and B. The factor e' - e de- 
pends on the temperature difference T' - T 
alone and is again independent of the shape 
of A and B. Whence it follows that the 

heat lost by A per unit area should be expressible by the 

formula W^(T'-T)p x . 

This formula, however, will not hold for the heat gained per 
unit area by B, since some molecules issuing from a certain part 
of the enclosing walls hit another part of these walls and not the 
body A. Only the total loss of heat of the whole surface of A 
will be equal to the gain of the whole surface of B. 

20. THE EFFECT OF ACCOMMODATION UPON HEAT CONDUCTION 

It appears from Knudsen's experimental findings that the 
amount of heat transferred from one to the other body and the 
coefficient e K calculated therefrom are smaller than their theo- 
retical values. The heat conduction depends on the relative 
dimensions of the surfaces and on their nature. For rough 
surfaces the experimental and the theoretical values of e K differ 
from each other less than for smooth ones. 

To account for this Knudsen considers the effect of accom- 
modation. 

Once more imagine the two plates S and S' (Fig. 8) at the 
temperatures T and T'. Let 6 and 0' be the temperatures of 
the gas molecules leaving 8 and S', and a and a' the accommoda- 
tion coefficients of these plates respectively. 



in INVESTIGATIONS ON RAREFIED GASES 141 

According to the definition of the accommodation coefficient 
(cf. p. 134) we have the relations 

0'-6~a(8'-T), 
0-e'=a'(6-T'), 
from which follows 



In the expression (42) for the heat transfer T - T has now to be 
replaced by 0' - 0, which can be accomplished by writing 



where 

- aa 



Knudsen points out that what is measured in the experiments 
is this new coefficient K . Among the special cases he considers 
first that in which both surfaces are of the same kind, so that 
a = a', the coefficient being then denoted by e u , and then the 
case in which the plate 8' is infinitely rough so that a' = 1 and 
the coefficient is denoted by e lx . The coefficients e K , U , and 
e lx are related to each other by the equations 

a 
* n ** 2 - a * Ky IQO = ajr * 

K being the value of the coefficient when both plates are infinitely 
rough, so that it might also be written <*> , . 

The magnitudes , and appear to form an arithmetical 
series. 6 * i n 

Knudsen determined these different magnitudes by measure- 
ments in which the heat transfer took place between two co- 
axial cylinders. It appeared that the heat transfer depended 
here on the distance of the surfaces and was, for a given pressure, 
greater when the cylinders differed considerably in their radii 
than when they surrounded each other closely. This can be 
explained by noticing that with a large distance between the 
surfaces of the cylinders the molecules which leave the heated 
inner cylinder C' do not return to C' after a single collision with 
the outer cylinder C, as would be the case for a small distance 



142 KINETICAL PROBLEMS CHAP. 

of the surfaces, but that between two 'collisions of a molecule 
with C' several collisions with C can take place, so that the 
molecule has more opportunities to give up heat to the inner 
side of the cylinder C. That the nature of the surface also has 
an effect upon the heat conduction will become plain by assuming 
that a colliding molecule gives up the more of its energy to a 
wall, the more rough the latter. In the case of a perfectly rough 
wall the molecules give up at a single collision their excess of 
heat, and the distance of the surfaces will, therefore, have no 
influence upon the rate of the heat conduction. 

We will consider, then, the two co-axial cylinders C' and (7, 
kept at the temperatures T' and T respectively, and in calculating 
the heat conduction we will follow Knudsen's not quite rigorous 
method. A perfectly rigorous treatment would offer too great 
difficulties. 

Let us assume that a molecule which flies away from C' 
returns to C' after n collisions with C. Further, let the tempera- 
ture of molecules on leaving C' be 0', after a single collision with 
(7, I9 after two collisions with C, 2 , and after n such collisions O n . 
If a is the accommodation coefficient for both surfaces, we have 
the relation 

f -fl 1 =a(0'-T) > 
whence 

0, - T - (1 - a)(0' - T) = 6(0' - T), 

where 6 = 1 - a. After n collisions with C we have 



and a collision with C' gives 

0'-r = 
From these equations we derive 

0'-0 a =a 
so that 



This expression gives us the amount of heat carried over from 
C' to C by molecules which between two collisions with C' collide 
n times with C. 



in INVESTIGATIONS ON RAREFIED GASES 143 

Knudsen splits the heat transfer into parts contributed by 
molecules which between two collisions with C' collide with C 
once, or twice, and so on. For this purpose, however, one has 
first to find the probability that a particle which flies away from 
the outer cylinder C will hit the inner cylinder C'. 

We have always assumed after Knudsen that the number of 
particles which are sent out by a surface-element in directions 
contained within a cone of aperture dco 9 and with an axis making 
an angle a with the normal to the surface-element, is proportional 
to cos arfw. If we now consider a surface-element placed at a 
point P of the outer cylinder, the required probability will be 
represented by 

w =/' cos 
/ 2 cos 

the integral 1 being taken over all cones which have their 

vertex at the chosen point P of the 

outer cylinder and which intersect the 

inner cylinder, while the integral 2 

has to be extended over all cones, 

with the same vertex P, within a p 

solid angle 2?r. We lay through P 

two planes (Fig. 11), PRA, passing 

through the common axis of the "^ F IO . 

cylinders, and PRB, touching the 

inner cylinder. Let QR = <f>, ^QRA=0, BRA=0. Then 




Since a is the spherical distance QA, cos a=sin < cos 0, and 
therefore, 



/ I " sin 2 </> cos 0d0d<f> 
^0 ><> 



where r is the radius of the cylinder C' and R that of C. Thus, 
of all molecules leaving C' a fraction r/R will again return to C' 
after a single collision with C. The remaining part 1 - r/R of 
the molecules will have collided with C more than once. Of 
these again the fraction r/R will return to C' after a second 



144 KINETICAL PROBLEMS CHAP. 

collision with C. Thus, the part of all molecules which return 

to C' after two collisions with C is (l-4)> an( * tlie P art 

B\ K 

rt r\ n ~ l 
returning after n collisions, - ( 1 - . ) 

R\ RJ 

The total heat transfer can now be represented by 

W = (T' 
where 



E 

while 



(43) 



This formula was tested by Knudsen in a series of experiments. 
He worked with two glass cylinders, of which the inner one 
consisted of a thin-walled tube around which many windings of 
platinum wire were coiled and fused into the glass. The radius 
of this tube was r = 0-340 cm. To the platinum coil two stouter 
platinum wires were attached by means of which the tube could 
be suspended within the wider glass cylinder. In the first 
measurements the radius of the outer cylinder was JRj =0465 cm., 
and in later ones 7? 2 = 1-61 cm. The outer cylinder is placed in 
melting ice, while the inner one is heated by an electric current 
passing through the platinum coil. The resistance of the latter 
and the intensity of the electric current are measured and from 
these the temperature of the inner cylinder and the amount of 
heat generated are calculated. Since the measurements are made 
after the stationary state is established, this amount of generated 
heat is equal to that of transferred heat. The loss of heat was 
due partly to radiation and partly to molecular conduction. By 
means of a system of pipettes equal small amounts of gas cduld 
be introduced. This did not change the heat loss due to radiation 
but only that due to conduction, so that from a series of readings 
of the pressure, the temperature, and the amount of heat 
generated the amount of transferred heat per degree of tem- 
perature difference, per unit of pressure and unit of area, i.e. 
the coefficient e ff R , could be computed. The gas chosen was 
hydrogen. 



in INVESTIGATIONS ON RAREFIED GASES 145 

Knudsen found for e r RI 1-87 . 10" 6 cal. and for e r Ut 
245 . 10"" cal. These results show that, in accordance with 
the theory, the heat conduction increases with the distance of 
the cylindrical surfaces. 

The ratio of e r RI and e r ^ contains only the accommodation 
coefficient as unknown quantity, so that the latter can be com- 
puted from ^*'. One finds a =0-26. 

*r,7? 8 

With this value of a the coefficient e A - can be calculated. Its 
value thus found is 11-1 . 10 ~ 6 , while the. formula (43) based on 
the kinetic theory gives e^ = ll-0 . 10~, a good agreement 
testifying to the correctness of the theory. 

Some further measurements undertaken by Knudsen with 
the purpose of determining the change of the accommodation 
coefficient with the temperature need not detain us here. 

With regard to the accommodation coefficient it may still be 
mentioned that from the theoretical standpoint it is not satis- 
factorily defined. In fact, the accommodation coefficient has a 
meaning only if the state of the gas is completely determined by 
the mean kinetic energy of the molecules, if, e.g., it is assumed 
that Maxwell's law holds for the velocities of the molecules before 
as well as after their collision with a solid wall. This is for the 
gas between the two cylindrical surfaces not quite the case. 

For a rigorous treatment of the question one would have to 
take into account, for a given temperature of the wall and a 
given velocity of the arriving molecules, the probability of a 
determined state of the molecule after it rebounds from the wall. 
This problem, however, is too intricate. 

21. HEAT CONDUCTION IN A GAS OP GREATER DENSITY 
CONTAINED BETWEEN TWO PLATES OF UNEQUAL TEMPERATURE 

After this treatment of the heat conduction in strongly 
rarefied gases we will now give a short account of thermal 
conduction in a gas of greater density. Whereas it is known 
that the coefficient of heat conduction for large gas densities is 
independent of the density, the intermediate domain between 
the very small densities, explored by Knudsen, and great 
densities requires still a detailed investigation. We have 
already touched this domain when treating of a streaming gas 



o 



146 KINETICAL PROBLEMS CHAP. 

layer,* and we have then started from large densities. It 
appeared that with decreasing density next to friction also 
sliding began to assert itself. We will now inquire into the 
behaviour of the heat conduction coefficient in this domain. 

We consider two plates kept at constant temperatures T and 
T'. Let the density of the gas be such that the mean free path 
of the molecules cannot be quite disregarded in comparison with 
the distance of the plates, although there are still very many 
collisions. The temperatures of the gas layers in contact with 
the plates will differ somewhat from T and T as was already 
pointed out by Kundt and Warburg. 

Fig. 12 gives a graphical representation of the temperatures 
of different layers. 00' is the distance of the plates, OA and 

Q'A' represent the temperatures T 
and T' of the plates, and OB and 
O'B' those of the adjoining gas 
layers as measured by the mean 
kinetic energy of the molecules. 
For large densities the temperature 
of the gas layers may be represented 
by the straight line AA'. 

At some distance from the fixed 

*~~~^AI plates the theory for large densities 

^ I holds, so that the temperature 

N// gradient is uniform and the tem- 
Fics. 12. perature line has there a straight 

portion. We produce the latter up 

to the intersection points D and D f with the vertical lines 
through A and A'. We assume that T -T is infinitesimal, 
in case the heat conduction coefficient should depend on tem- 
perature. This condition enables us to assume that if the 
temperatures of the plates are increased or diminished by equal 
amounts, the line does not change its shape. 

The whole figure can thus be shifted horizontally without 
changing the length of the lines AD and A'D'. Again, the 
figure can be transformed by increasing or diminishing all the 
distances from AQ in the same ratio. In this case also the 
lengths of AD and A'D' remain unchanged ; they are thus inde- 
pendent of the temperatures of the plates. If both plates are of 
* Cf. Chapter II. Art. 10. 



in INVESTIGATIONS ON EAEEPIED GASES 147 

the same kind, we should find the same curve by interchanging 
their temperatures. This curve will be found by taking the 
mirror image of the original one with respect to AQ. Whence it 
follows that the parts of the curve at B and B' are congruent, 
and that therefore AD=A'D'. 

Moreover, since the curve retains its shape when the distance 
of the plates is changed,* the length AD=A'D' for a given gas 
will depend only upon the nature of the plates and will be 
independent of their temperature and their distance. 

Let the distance of the plates be I and AD =A'D f = A. 

Similarly to what was said on p. 104 about the significance of 
00 and BD in Fig. 2, a physical meaning can be ascribed to A. 
In fact, if the gas extended also outside of the plates and had 
throughout the same temperature gradient as actually exists 
within the gas, the temperatures at the points D and D' would 
be equal to those of the plates, i.e. T and T' respectively. 

The temperature gradient in the gas at not too small distances 

f T T" T 

from the plates is then and the heat transfer is k . 

1 Z + 2A Z + 2A 

where k denotes the conduction coefficient. 

Whence it follows that if the plate distance I is increased the 
amount of heat transferred does not vary inversely proportionally 
to I but slower. 

We will now prove that, for given temperatures of the plates, 
A is inversely proportional to the density while k is independent 
of the latter. 

For this purpose we compare two geometrically similar cases 
[systems] in one of which the gas density is n times greater and 
the dimensions n times smaller than in the other. Let P t and 
P 2 be two corresponding points, and let the state at these points, 
apart from the density, be exactly the same. Let ds be a volume 
element at P l and FdsdX the number of molecules contained in 
it, whose velocity components and quantities determining jbhe 
inner state fall within a given domain d\. A similar group in an 

equal element at P 2 will then contain -- Fdsd\ molecules. We 

n 

* That AD~A.'D' is independent of the distance of the plates can be seen 
by inserting between these a third plate P of such a temperature as not to 
change the state of the gas apart from the immediate neighbourhood of P, 
and by applying a similar reasoning to that used on p. 103 to prove that the 
sliding coefficient v is independent of the distance of the plates. 



148 KINETICAL PROBLEMS CHAP. 

assume that the temperatures of the plates are the same in both 
cases ; then the temperatures of gas layers at corresponding 
points will also be equal. This harmonises with the assumption 
that at corresponding points the number of the molecules in 
different groups in one gas is the same fraction of their number 
in the other gas ; therefore the averages of all quantities are 
equal, and this holds then also for the temperature as the mean 
energy. 

We will first of all prove that the possibility of the existence 
of the first state carries with it also that of the second. We 
consider in the first state a group of particles and follow their 
history during a certain time r. The parameters characterising 
these particles are contained within a given domain, while the 
particles themselves lie in a volume element ds. If there were 
no collisions, we would find these particles at the end of the time 
T in a volume element <fe'. Owing to the collisions some particles 
will leave the group and others will join it. The condition that 
the state should be stationary implies that there are in (Is' at the 
beginning and at the end of the time r equal numbers of particles 
whose velocities and parameters determining the inner state fall 
within certain limits. We assume such to be the case for the first 
state and shall prove that this holds then also for the second state. 

The quantities concerning the first state will be distinguished 
by the suffix 1 and those relating to the second state by the 
suffix 2. 

We consider in both cases a group of particles with equal 
intervals for the magnitudes characterising them and contained 
within equal volume elements (ds)j, = (ds) 2 placed at corresponding 
points. We take the time interval r for the second system n 
times as long. Then also (ds'^ and (ds') 2 will lie at corresponding 
points. If collisions are disregarded, the number of particles 
contained in (ds\ at the beginning of that time interval will be 
n times that contained in (ds') 2 . This then will also be the case 
at the end of the time intervals T X and T 2 , since all these particles 
arrive from (ds^ and (ds) 2 . If there are collisions, then things 
are not so simple. It may be noticed, however, that the number of 
particles which leave the group owing to collisions and the number 
of those that join it for the same reason amount in both systems 
to the same fraction of the total number of particles. In order 
to see this, the collisions may be classified according to circum- 



in INVESTIGATIONS ON RAREFIED GASES 149 

stances. Under coinciding circumstances the number of particles 
in the first system is n times, and, therefore, the number of 
collisions n 2 times as large as in the second system. But since 
the time interval for the second system was taken n times as 
large, the number of particles leaving or joining the group in the 
first system will not be n 2 but only n times as large. The ratio 
of densities will thus remain in both systems the same ; also 
with collisions will (<fe')i contain n times as many molecules 
as (ds') 2 . 

For the interior of the gas it is thus proved that if the state 
of the first system is stationary, so is that of the second. And 
if we further assume that the particles rebound from both plates 
in the same way, this theorem will hold also for the limiting gas 
layers at the plates. 

Lastly, to prove that A is inversely proportional to the 
density, we note that the graphical representation of the tem- 
perature distribution for the second case can be simply deduced 
from that of the first. In fact, since we have assumed that the 
respective temperatures of the plates are equal in both cases and 
that this is true also of the temperatures of gas layers at corre- 
sponding points, the horizontal dimensions of the two figures are 
equal, while the vertical ones of the second are n times those 
of the first. Whence it follows that A^wA^ which proves 
that A is inversely proportional to the density of the gas. It 
remains only to prove that the conduction coefficient k is 
independent of the density. 

22. HEAT CONDUCTION COEFFICIENT INDEPENDENT 
OF DENSITY 

In order to prove this, we consider the number of molecules 
which pass in equal times through unit area of corresponding 
planes V t and V 2 parallel to the plates, and we compare again 
similar groups of molecules. 

This number is for the first case n times as great as that for 
the second, and this holds then also for the total energy carried 
across the surface-element. Consequently, the heat conduction 
in the first case is n times that of the second, and since the 
temperature gradient is also n times as great, the coefficient k 
will be the same for both cases. 



150 KINETICAL PROBLEMS CHAP, m 

Moreover, noticing that A and k do not change when at 
constant density of the gas the plates are moved farther apart 
or brought nearer to each other, we can say, generally, that A is 
inversely proportional to the density and that k is independent 
of density. 



23. LASAREFF'S EXPERIMENTAL INVESTIGATION 

Lasareff * investigated the temperature distribution in the 
immediate neighbourhood of a wall. He took a very thin gas 
layer, about 9 mm. thick, and worked with highly rarefied gases. 
As walls, metal plates were used which were kept at constant 

_. _ __ temperatures by means of 

B JJ f 12 . . -, -, . i . 

water jackets, a cold jacket 
being placed under the lower 
and a warm one above the 
upper plate. The tempera- 
tures were measured by means 
of a thermo-electric pile of 
which one junction was placed 
near the cold plate and the 




FIG. 13. G other could be moved up and 

down in the space between 

the plates. Fig. 13 gives a graphic representation of Lasareff's 
experimental results. The abscissae represent the temperature 
difference relatively to the cold plate and the ordinates the 
distance from the heated plate. The curves AB, CD, EF, and 
GH give the observed results at a pressure of 760, 0-087, 0-065, 
arid 0*019 mm. mercury respectively. 
Lasaxeff found for hydrogen : 

p (mm. mercury) 7 

4-5 0-022 

2 0-055 

7 is a coefficient, proportional to A. Inasmuch as it can 
be assumed that the temperature of the wires of the thermo-pile 
actually coincides with that of the gas, these experimental results 
prove satisfactorily that A is inversely proportional to the 
density. 

* Lasareff, Ann. der Phys., xxxvii., 1912, p. 233. 



CHAPTER IV 
REMARKS ON LESAGE'S THEORY OF GRAVITATION 

24. LESAGE'S THEORY OF GRAVITATION 

KNUDSEN'S investigations on rarefied gases may be connected 
with the old theory of gravity due to Lesage.* According to this 
theory celestial space is full of smaJl particles or corpuscles 
moving in all directions with great velocities. Material bodies 
are incessantly hit by these corpuscles and throw them back. A 




FIG. 14. 

body placed alone in space will be hit in all directions by equal 
numbers of particles and will thus experience no resultant effect. 
Two bodies A and JB, however, would partly shield each other 
from the impact of the corpuscles and consequently appear to 
exert upon each other an attractive force (Fig. 14). If the 
corpuscles are assumed to be small compared with the atoms, 
this would lead to an attractive force between two atoms. 

Since the number of corpuscles intercepted by one atom is 
proportional to the solid angle under which it is seen from the 

* Journal des savants, 1764. 
151 



152 



KINETICAL PROBLEMS 



CHAP. 



other atom, the force would be inversely proportional to the 
square of distance of the two atoms. 

But it does not follow from the theory that the force is 
proportional to the masses ; it would, instead, depend on the 
dimensions of the bodies. To secure the proportionality of the 
force to mass, yet another hypothesis would have to be introduced. 

Meanwhile the theory of Lesage was shown by Maxwell to be 
incorrect. The motion of the corpuscles is much the same as 
that of gas particles, and as against the fact that the body B 
intercepts corpuscles which in its absence would have reached 
the body A there is this other fact, that due to reflection from B 
some corpuscles will reach A which otherwise would not do so. 





FIG. If). 



Fin. 1C. 



It is not possible to keep the space between two bodies free from 
corpuscles, no more than to keep a space free from black body 
radiation, even if the mean free path of the corpuscles is great 
and their mutual collisions may be disregarded. In the case of 
two parallel plane plates, for instance, only those corpuscles 
which hit the plates perpendicularly would not reach the space 
between the plates ; but this is only an infinitesimal fraction of 
the total number of corpuscles. 

It would be difficult to prove that a space can never remain 
free from corpuscles. In fact, one can very well imagine a 
particular state of motion for which a certain domain remains 
free from corpuscles, if, e.g., these move originally outside a 
certain sphere B and inside a second sphere B 2 concentric with B. 
If the surface of J? 2 is perfectly reflecting and if there are no 
collisions between the corpuscles, then the corpuscles will never 
penetrate into the inner sphere (Fig. 16). But such a state of 
motion will never arise. 



iv LEPAGE'S THEOKY UF UKAV1TAT1OJN 153 

The theory of Lesage can be saved by assuming that the 
corpuscles are wholly or partially absorbed by matter. But then 
the picture is deprived of its simplicity. 



25. AN ELECTROMAGNETIC ANALOGUE OP LESAGE'S THEORY 

It was once asked whether Lesage's theory can be given an 
electromagnetic form. One would then have to assume, for 
instance, that space is full of radiation of a wave-length much 
shorter than that of Rontgen-rays and to show that two particles 
would be driven towards each other by the radiation pressure. 
Such would, in fact, be the case if the particles continually 
absorbed the radiation. 

Consider two electrically charged particles P and Q, at a 
mutual distance r, in a space traversed by electromagnetic 
waves. Let E be the electric, // the magnetic force, and n the 
frequency. Evidently the particle P will be set into vibrations, 
whether it is free or bound to a position of equilibrium ; only 
the type of vibrations will be different in the two cases. Owing 
to the radiation emitted from Q the radiation field in which P 
is placed will be modified. 

Let us introduce a system of co-ordinates with P as origin, 
and let the co-ordinates of Q be -r, 0, 0. 

In a first approximation the displacement components of P 
can be written 



z=aeE z -beE z . 

This is a convenient way of expressing the phase difference 
between the incident waves and the oscillation produced by them. 
The 'phase difference is represented by the second terms and is 
caused by the resistance experienced by the particle due to 
friction or emission of radiation. The coefficients a and 6 are 
supposed to be constants which depend on the mass of P, on the 
restituting force, etc. In E x , E y9 E z is included the field which 
the particle Q, being also set vibrating, produces around itself by 
its radiation. 

Noticing that the work done by the electric force against the 



154 KINETICAL PROBLEMS CHAP. 



resistance, i.e. eEyti+eEyfi+eE^, for a full period, is positive, it 
will be seen that b must be positive. 

As soon as P is set vibrating, the force upon it is no longer 
E as at the origin [position of equilibrium]. The ^-component 
of the force can now be written, in a second approximation, 

*(yH z -zH y ) +e (x--- 

c \y z JJ \ dx y dy dz 

where for H yi H gi E x and their derivatives are to be taken the 
values belonging to the original, i.e. the equilibrium position of P. 
The first term is due to the velocity of the particle, and the second 
due to its deviation x, y, z from the position of equilibrium. 
These terms give rise to very weak oscillations with twice the 
frequency of the incident waves. 

Now, to find the as-component of the force acting upon P, the 
last expression has to be averaged over a full period. The 
required, rather lengthy calculations, in which account must also 
be taken of the disturbance of the field by the vibrations of Q, 
may here be omitted. It appears that, if terms with 1/r 3 , etc., 

be neglected (which is permitted provided r is large compared 

P 
with the wave-length), the term (ifH z -zH y ) alone survives, and 

c 

this also inasmuch only as it depends on the terms -beE in 
the preceding equations. 

As a result of these calculations one finds for the required 
force 

-b e ^(E y H z -$JBj ^ e \E y H z -E,H U ), 
c c 

where for E and H is to be taken the field as it is in presence of 
Q, but neglecting the effect of P upon it. 

This result can be associated with an energy flux. In fact, 
the expression c(E y H z -E z H y ) represents the s&-component of -the 
energy flux at P. Since we assume that the rays are propagated 
equally in all directions, there will be in all points of a sphere 
with Q as centre and of radius r the same radial energy flux. If 
E be the flux of energy through the whole spherical surface, 
reckoned positive when directed outwards, then 

c(E y H z - 



iv LESAGE'S THEORY OF GRAVITATION 155 

The component of the force upon P taken along the X-axis 
will thus be represented by 

E 



Since Q is placed on^the negative X-axis, this expression will 
represent an attractive force if it is negative, that is to say, if 
E is negative. Such will be the case if more energy streams 
through the sphere inwards than outwards, and therefore, if Q 
absorbs the rays. 

The electromagnetic modification of Lesage's theory leads 
thus to a similar result as the original corpuscular theory. 



CHAPTER V 
FRICTION AND HEAT CONDUCTION IN THE PROPAGATION OF SOUND 

26. THE EFFECT OF FRICTION 

WE will now consider the effect of inner friction and of heat 
conduction upon the propagation of sound in a gas. 

Let u, v, w be the velocity components of a volume element 
and $ the relative condensation, i.e. if p be the actual density 
and p that in the original state of equilibrium, s = (p-p Q )/p Q . 

Further, let + CV - + ~. -=K, and let us for the present disregard 

the heat conduction. Then the equations of motion for the 
propagation of sound waves will be 

ff + S *-0 (44) 

/v ^ s* v v"/ 



ds . dv / A ..*.,. i ^ 



2 ds dw _ 

Here a would be the velocity of propagation of sound in absence 
of inner friction, and v =p/p, where p. is the friction coefficient, or 
the viscosity.* 

* Formula (44) is the equation of continuity for a compressible fluid, viz. 



, . 

It ox cy dz ~~ ' 

in which the terms u *-, etc., are neglected, the deviations from equilibrium 

being assumed infinitesimal, and the term - ^ is replaced by - ~ = fy' 

156 



CHAP, v PROPAGATION OF SOUND 157 

Differentiating the three equations (45) with respect to z, y y z 
and taking account of (44) we have an equation for s 9 

9A d 2 s 4 9A A fAC >. 

a2As -^V fo = (46) 

This is, for v=0, the well-known equation of propagation of 
sound waves. 

The problem of propagation in an unlimited gas mass can be 
easily solved for the case of plane waves. 



The equations (45) follow from the equations of motion 

. . (a) 



ex vy vzj <x cy *cz 
(cf. p. 77). 

The expressions for the stress components can bo found by following the 
hints given in the footnote to page 78, with the only difference that our fluid is 

now compressible so that ^ f ^ + ^- -K (locs :iot; vaili h 
Thus one finds 

X* = -P + Ki -OM)' 



and similar formulae with the same coefficients a u and 2? for the remaining 
stress components. In these equations p is the pressure, as it would correspond 
to the density and temperature at the given point if the gas were at rest or if 
it had throughout the same velocity w, v, w. 

Between a n and a 2a holds, moreover, the relation (<i u -a 22 ) + 3a 22 = 0, which 
can bo deduced by going deeper into the manner of arising of the stresses, but 
which can be accepted here without proof. 

Ultimately, putting a n -a 32 =2 / u, we find 

m 2 , r 



and substituting these expressions for the stress components in (a), 

^f,ctc. . . (b) 



u = - 

ex ?y czf C'X 6 

Assuming Poisson's law p=C(P, wo have 



P 
Again, =a a , while - can be replaced by - , so that 







This, substituted in (6), gives the equations (45), if we put pfp^v and 
neglect products of velocities into their derivatives. 



158 KINETICAL PROBLEMS CHAP. 

If we put 8 =$06^"**, we can find at once for every frequency 
the corresponding value of q and thence the propagation velocity 
and the damping of sound. 

We will consider here the propagation in a gas contained 
in a cylindrical tube, whose axis coincides with the a-axis. The 
propagation is to proceed along the ovaxis, so that all terms will 
contain the factor e int ~ qx . 

Since ~ =ins and ~ = -w 2 *, equation (46) becomes 
ot vt A 

(a 2 + *i/wi)A*+n 2 $=0 (47) 

Since the state in a cylindrical tube is symmetrical about its 
axis, we introduce, instead of y and z, the distance r from the 
axis, so that 

8 2 3 2 3 2 19 



Again, = ? 2 s, and therefore, instead of (47), 
8a; 2 

d z s I ds 
8r 2 r dr 

This is the differential equation of the Bessel function 7 , so that 
we have, for the condensation, 



, .... (48) 

2 



where A 2 - /- 

(a? 

The argument of / is thus a complex number. 

The value of u is now to be found from the first of equations (45), 

where K = - = - ins, so that this equation reduces to 

ot 

rv 

q(a 2 + livri)s ..... (49) 



Of this a particular solution can be found by putting u = f and 
by suitably choosing the constant . Substituting this in the 
left-hand member of (49), we have 



(ins -vks), 
so that $(in8-v&8)=*q(a 2 + livn)8. . . . (50) 



v PBOPAGATION OP SOUND 159 

Since, by (47), ___ n_ 



we see that equation (50) is satisfied by 



To this particular solution for u an arbitrary solution of the 
equation 1 - j/Aw = can still be added. 



Since u is a function of x and r only, the last equation can 
be written 



d z u lou ( , in\ A 

-^- + -x- + (q w=0. 

cir T or \ v / 

and this is again satisfied by a Bessel function, 

u = c 2 e 7W *~" Qa: / (JBr), where J3 2 = q 2 - 

p 

Thus the solution for u becomes 

. . . (51) 



It remains only to determine the velocity components v and w. 
Owing to the symmetry the velocity in a cross-section of the 
tube will be radial, so that v and w can be represented by v=yh 
and w=zh, where h will, apart from the factor e (int ~ qx \ depend 
on r alone. If these values be substituted in the second and the 
third of equations (45), all terms in the second equation will have 
the factor yfr and all terms in the third the factor z/r. On being 
divided by y and z respectively, these equations turn out to be 
identical and after a multiplication by r each of them reduces to 



O =0. . (52) 

The third terra of (52) can be evaluated by means of the equation 
of continuity (44) which can be written 



dv 



160 KINETICAL PROBLEMS CHAP. 

Substituting here v=yh and w=zh and differentiating with 
respect to r, we have 

3A 3% . ds 



and this introduced into (52) gives 

/ \ -L i * * \3* 
- (iw - vq*)rh = (a 2 + &nv)^ 

Since ~ and ' follow from the solutions (48) and (51) for s and 

dr dr 

u, the radial velocity rh is herewith determined. 

To determine the constants o t and c 2 , we have two boundary 
conditions. For greater densities, when there is no sliding, the 
gas is at rest at the walls, so that for r = R (radius of the cross- 
section of the tube), u = and rh = 0. This gives the equations 



- qvc 2 BI '(BR) = 0, 

where 7 ' is written for the derivative of J . 

Eliminating from these two equations c^c^, we find 



Here q is the only unknown. Since q is contained also in A and 
B, its determination from this equation is very laborious. One 
has to use approximation formulae for the Bessel function and to 
assume that the effect of friction is small and thus also that 
the state differs but little from the propagation in a non- viscous 
gas. 

27. EFFECT OF HEAT CONDUCTION 

This problem was treated by Kirchhoff, who has taken into 
account also the influence of the heat conduction. 

In deducing the equations of motion (45) use was made of 
Poisson's law for adiabatic volume changes (cf. footnote on 
p. 157). If, however, heat conduction is taken into account, 
Poisson's law can no longer be applied, and one has to write 
down the thermical equation which concerns the change of the 
internal energy of a volume element due to compression and 
heat conduction. 



v PKOPAGATION OF SOUND 161 

Kirchhoff * finds for the propagation velocity in a cylindrical 
bube of diameter 2/2 

> .... (53) 



where n is the number of oscillations per second and a the pro- 
pagation velocity in an unlimited three-dimensional space, in 
which case the effect of friction is small ; 7 is a constant 
depending on viscosity and heat conduction, viz. 



^viscosity 
density ' 

7 coefficient of heat conduction 

K = r rr 3 

density 
while b is the propagation velocity as calculated by Newton, so 

that?- As. 
b Ve; 

Formula (53) was repeatedly tested experimentally but was 
never found well corroborated. Kayser f found the deviation of 
the propagation velocity from the value a about four times as 
large as that required by the theory. 

In a space of three dimensions everything becomes much 
simpler than in a tube. The effect of viscosity is then the greater 
the smaller the wave-length. 

Neklepajev J investigated, in connection with Lebedew's theo- 
retical researches, the propagation of very short sound waves in 
air. His source of sound was an electric spark produced at the 
focus of a concave mirror Sj. The beam of parallel rays (for 
such short waves, as e.g. 0-2 cm., one can speak of " sound 
rays ") reflected by the mirror S x fell upon a diffraction grating 
consisting of a series of silvered steel rods. Diffracted bundles 
were thus produced, and one such bundle was concentrated by 
a second concave mirror S 2 upon a sensitive vane which was 

* Pogg. Ann., cxxxiv., 1868, p. 177 ; Ges. Abh. t Leipzig, 1882, p. 540. 

t Wied. Ann., ii., 1877, p. 218. 

J Ann. der Phys., xxxv., 1911, p. 175. 

Ibid., xxxv., 1911, p. 171. 

VOL. I M 



162 KINETICAL PROBLEMS CHAP, v 

displaced by the pressure of the sound rays. The wave-lengths 
were measured by means of the grating, through the diffraction 
angle. Neklepajev worked with wave-lengths of 25 down to 
0*85 mm. 

By sending the sound rays through layers of air of different 
thickness the absorption could be measured. This appeared to 
be considerable. The experiments gave a higher value for the 
absorption than was to be expected from Lebedew's theoretical 
considerations. Lebedew gives for the distance in which the 
intensity is reduced to (-J ff of its original value, for different 
wave-lengths, the following figures : 

X in mm. Distance in cm. 

0-8 40 

0-4 10 
0-2 2-5 

0-1 0-6 

For polyatomic gases, in addition to v and k yet a third 
coefficient must be introduced. In fact, when the temperature 
rises while a volume element is being compressed, this will increase 
the velocity of the translational motion of the molecules as well 
as the intensity of their inner motion. In the state of equilibrium 
the energy of translational motion bears a determined ratio to 
that of the inner motion, but while the velocity of translation is 
directly affected by the compression of a volume element, the 
effect upon the velocity of the inner motion is not so immediate. 
The internal energy remains thus in its fluctuations, so to speak, 
behind the energy of translation. This gives a coefficient affecting 
the propagation velocity of sound, viz. making it somewhat 
smaller. It remains to be seen, however, whether this coefficient 
can have a perceptible value. 



CHAPTER VI 

KINETIC THEORY OF SYSTEMS OP ELECTRONS 
RICHARDSON'S INVESTIGATIONS 

28. THEORETICAL INTRODUCTION 

RICHARDSON* has made an important investigation on the 
emission of negative electrons by a hot metal, and found that 
this is due to the heat motion. The electrons, endowed with 
great velocities, will escape in spite of the forces exerted by the 
metal. The escaping electrons were found by Richardson to 
have a kinetic energy agreeing with that of gas molecules. 

We will, first of all, follow here Maxwell's considerations on 
the velocity distribution in a monatomic gas acted upon by an 
external force, as e.g. the gravity. 

Let % , rj t be the velocity components and x, y, z the co- 
ordinates of a gas molecule. We consider the molecules whose 
velocity components and co-ordinates are contained between the 
limits f and f + d% , ij and 77 + d rj, and % + d%,x and x+dx,y and 
y+dy,z and z + dz. Their number will be represented by 

fdgdr)ddxdydz =fds. 

The components of the external force will be denoted by X, Y, 
and Z. 

AJ1 this being valid for the instant t, let us now follow the 
history of this particular group of particles. Mutual collisions 
being excluded, these particles will be contained at the instant 
t+dt within a phase element ds' placed at a point of the six- 
dimensional space whose co-ordinates are 

+ ^cft, !? + -<&, + -&, x + gdt, y+vdt, z + dt. 
b m 'mm > * f > * 

* Phil Mag. (6), xvi., 1908, p. 353. 
163 



164 KINETICAL PROBLEMS CHAP. 

Let the state of the gas be stationary. This implies that the 
number of particles which fall within the phase element ds' is 
the same at the instant t as at the instant t +dt y so that 



. .)d'=fds. 
But by Liouville's theorem ds' =ds, so that 

/(*+*' V+#*C%^ 

or, since the difference of these functions is zero, 



dfX dfY dfZ fif. a/ 

4 - +^ +4-+/-^- 

0w 077 w df w PX dy 
This equation can be satisfied by 



where a is a function of the co-ordinates and h is independent of 
them. 

In fact, on substituting we find (with v 2 = f 2 + ^ 2 + f 2 ) 



3o a 9A\ ../So ,Sh 



an equation which must 'be satisfied for all values of f, y, f, 
x, y, z, so that the coefficients of f, f 3 , etc., must all vanish. 
This gives 



Thus, the external force must have a potential. If ^ be the 
corresponding potential energy, 



and 

" 



vi KINETIC THEORY OF SYSTEMS OP ELECTRONS 165 



Since /=a6"^'* fl?i+ ^ ) , we find in the well-known way 3/2A for 

\3/2 



O 



the mean square of the velocity and a[ -) for N, the number 

\*/ 
of particles per unit volume. 

This can be written A r =^V e~ w x , if N Q be the number of 
particles at such places at which the potential of the external 
force is nil. This shows that the density of particles is smallest 
where the potential energy is greatest. 

Since in what precedes no assumption was made about the 
peculiarities of the field of force, apart from the existence of a 
potential, the results arrived at may be utilised in our further 
considerations on the motion of electrons. 

We begin with a limiting case in which the external force 
acts only within a thin layer and is directed normally to the 
boundary of the two media. Let Xi v 

be the potential energy in the first, 

and 2 th 8 ^ m *he second medium. _2 

Both magnitudes are constant, while 

the potential energy changes discon- 

tinuously across the boundary. If 

the medium 1 be a metal, and the FIG. 17. 

medium 2 the space above it, then the 

velocity distribution of the particles in both media obeys 

Maxwell's formula, the value of h being the same for both. 

Thus the densities will be given by 



Owing to the force in the boundary layer the velocity of the 
particles will be diminished in traversing this layer. In spite 
of this the mean velocity of the particles will be the same in 
both media, the reason being that only those particles leave the 
first medium which have the greatest velocity. 

To test this result by an explicit treatment, let us intro- 
duce a co-ordinate system of which the yz-plane coincides with 
the boundary of the two media and the positive #-axis extends 
into the second medium. 

We consider a particle [electron] which at the instant t is 
contained in the medium 1 and whose co-ordinates, x, y, z, % , rj, 
fall within the phase element ds. Let x', y', z' y %', rj', ' be its 



166 



KINETICAL PROBLEMS 



CHAP. 



co-ordinates at an instant t + T, where r is a finite time interval, 
long enough for the particle to have crossed the boundary at the 
instant t + r. It will be assumed that it does not collide, with 
atoms or with other electrons. 

The condition that the sum of the potential and the kinetic 
energy should be constant gives the equations 



'= 



. 



(54) 



Again, noticing that the time required to reach the boundary is 
-aj/f, we find the relations 



. (55) 



With the aid of (54) and (55) it can be easily verified that the 
state is stationary. In fact, at the instant t there are/<fa particles 
within the phase element ds and these will have passed at the 
instant t +r into the phase element ds 1 , while the latter contained 
f'ds' particles at the instant t. The condition for a stationary 
state will thus be 



Now it can be readily shown that ds = efe' and/=/'. In fact, 



~ds 



fit 





. ^000 


o 


S' 

-A o o 


o 


fc 











rV 
82 



tte 



vi KINETIC THEORY OP SYSTEMS OF ELECTRONS 167 
Again, 



and by (54), taking account of the values of a- L and 2 , these two 
expressions will be seen to be equal to each other. 

We now ask how many electrons pass, in the state under 
consideration, from 1 to 2. We determine first the number of 
particles which, per unit time, pass through a surface-element do- 
normal to the sc-axis in the first medium, and whose velocity 
components are contained between f and % + dg, rj and rj+dq, 
and f +d For this number we find ^fd^drjd^da. Similarly, 
%'f'dg'dri'dlZ'da' will be the number of particles which pass, per 
unit time, through a surface-element da' normal to the sp-axis in 
the second medium, and whose velocity components fall within 
the limits f and ' + d', rj' and rj' +<fy'> f and ?' +d. 

Let us now place da and da' in the immediate neighbourhood 
of the boundary and assume that f ', 77', f belong to , 77, f, i.e. 
that f ', 77', % are the velocities, after traversing the boundary, of 
a particle which in the first medium had the velocities (f , 77, f. 

It follows then from what precedes that 



If, therefore, da = da' 9 then also 



Herewith is also given the number of electrons which escape 
from the metal. If the state of the first medium is stationary, 
the number of electrons leaving it, per unit time, will be constant, 
even if the state of the second medium is not stationary, which 
is e.g. the case when this medium is unlimited so that the electrons 
are not thrown back. 

At the beginning of the heating the metal plate sends out also 
positive ions, which emission ceases, however, after a certain time. 
These ions come probably from a layer of gas condensed at the 
surface of the plate. 

29. VALIDITY OF MAXWELL'S DISTRIBUTION LAW FOR 
THE FREE ELECTRONS IN A METAL 

The ratio ejm for the negative particles emitted by the plate 
was determined by J. J. Thomson,* who was able to ascertain 

* Phil Mag. (6), xlviii., 1899, p. 547. 



168 KINETICAL PROBLEMS CHAP. 

that its value was the same as that known to belong to the 
electrons. Thus we have here the same corpuscles. Richardson 
measured their velocity, the arrangement of his experiment being 
as follows. 

Into a quadrangular aperture cut out in a platinum plate 
P (Pig. 18) was fitted the protruding part of the bent platinum 
strip /S, the latter being insulated from the plate P by mica. 

p The strip S was heated by an 

2 electric current. Another metal 
plate P 2 , connected with an electro- 

p meter, formed with the plate P a 

r\ condenser. The plate P 2 gathered 

F IG . is. the electrons escaping from the 

heated strip S. 

Richardson measured the potential of the plate P 2 during the 
process of charging, taking at the same time the utmost care that 
the centre of the platinum strip, the glowing spot from which the 
electrons were emitted, should 
remain at the potential zero. 

The velocity with which the 
potential of the upper plate 
mounts in its negative value 
will gradually decrease, because 
owing to the electric field thus 
produced not all the electrons 
leaving the strip S can reach 
the upper plate (Fig. 19). 

Let us first assume that all Flo 19 

electrons leave the strip with 

the same velocity u. There is then a maximum value <J> m to 
the potential of the upper plate. This will be attained when 
the electrons which move normally to the plates are just able 
to reach the upper plate. This gives the condition 



whence 



We can also find how the potential of the upper plate gradually 
increases. We will assume that the electrons which at the 




vi KINETIC THEORY OF SYSTEMS OF ELECTRONS 169 

potential $ just reach the plate have left the heated strip at an 
inclination 9 to the normal. These electrons describe then a 
parabola whose apex lies at the upper plate (Fig. 19), and we 
have, for 4>, 

iwH 2 cos 2 = e<D ..... (56) 

Let now n be the total number of electrons which, per unit time, 
leave the heated strip, and therefore 2n sin 9 cos Odd the number 
of those among them which move in a direction contained between 
9 and ff + d0 (cf . p. 128). Thus, the number of electrons for which 
6 is smaller than 6 is 



or, in virtue of (56), 



For <I>=0 this becomes equal to w, which means that then all 
electrons reach the upper plate. 

If C be the capacity of the condenser, the charge acquired by 
the plate P 2 , per unit time, is 

n d /- 2eO\ 
C - - =ew 1- J, 
dt \ mu 2 / 

whence, 



For t = oo this gives 4> w , while for t =0 we have assumed 4> =0. 
The intensity of the current is 

. dO - 2ne ' t 
i=C ,=nee "*. 
dt 

Thus, from the measured value of <i> m we can determine mu 2 /2e, 
and from the rate of increase of the potential the value of 
2ne*/mCu 2 . From these two magnitudes ne can be calculated 
and, since e/m is known, also the velocity u can be determined. 

We have thus far assumed that all the corpuscles leave the 
hot strip with the same velocity. Let us now see how the above 
result is modified if the velocities are distributed according to 
Maxwell's formula. It will appear that under these circum- 
stances the potential of the upper plate does not attain a 



170 KINETICAL PROBLEMS CHAP. 

maximum. In fact, when the potential of the plate has risen so 
high that the corpuscles of mean velocity do not reach it any 
more, corpuscles endowed with a much higher velocity will reach 
the plate and charge it to a yet higher potential. 

Let now the upper plate be a circular disc, and let us assume 
the field between the plates to be homogeneous, the potential 
difference of the plates being <1>. We have to calculate how many 
of the corpuscles leaving an element of the heated strip placed 
at a point will reach an arbitrary element of the upper plate. 
In doing so we shall assume the heated element to be very small 
compared with the receiving plate, so that all electrons which 
can overcome the potential difference will also be picked up by 
that plate. Let a be the distance of the two plates. If , 17, f 
are the components of the initial velocity of an electron, the 
number of electrons which, per unit time, leave the lower 
plate with velocity components contained between f and 
o> V an d y+drj, and +df, will be represented by 



The coefficient a will be determined by integrating over f from 
to oo and over rj and f from - oo to +00 and equating the 
integral to w, the total number of corpuscles, which, per unit 
time, leave the element of the hot strip. This gives 

a = ^ h 2 n. 

IT 

The integrated equation of motion of a corpuscle which left the 
glowing element (#=0) at the instant 2=0 with a velocity f in 
the direction of the -ST-axis is 



This gives, for x=a, the time t required by the corpuscle to 
reach the upper plate. If the corpuscle actually reaches the 
plate, this equation has two real roots, the smaller of which is 
the required value of t. 

We solve for I/t, and have therefore to take the greater of 
the two roots. Thus, 



t 2a V^ 2 2wa 2 ' 
This expression will be denoted by g. 



vi KINETIC THEORY OF SYSTEMS OF ELECTRONS 171 

Let now a rectangular element of the upper plate be hit 
whose sides are dy, dz and which is placed at a point y, z. Since 
y = ^, z = f& or y^gy and ?=<7z, the element will be reached 
provided that rj lies between gy and g(y+dy), and f between 
gz and <7(z + <fe). 

The number of corpuscles which leave the heated strip with 
the JK-component of velocity contained between f and f + d% 
and reach the given surface-element of the upper plate will thus be 



and the total number v of corpuscles hitting that element will 
be found by integrating this expression over the values of ff for 
which g is real. Thus, 

. . (57) 



This, integrated for y and z over the upper plate, gives the 
required number of electrons as a function of 4>. Multiplying this 
number by the charge e of an electron we shall have the current 
charging the upper plate, that is, Cd$>/dt. Richardson calls this 
the thermionic current. 

In Richardson's experiments the second [upper] plate was so 
large as compared with the mutual distance of the two plates that 
it may be considered as infinitely extended. This facilitates the 
integration of (57). Inverting the order of integrations we find 



/" fQ 
I 

/ 00 



(X7T 1 
" "A" 2A 



= 7be =ne 



where s = V2e<l>lm and n is the total number of electrons leaving, 
per unit time, the hot strip. 

The differential equation for the potential of the upper plate 

thus becomes j<D -^* A 



dt* 6 

Its solution is 

6^=1+2^, .... (60) 



where we have assumed <l> = for t =0. 



172 KINETICAL PROBLEMS CHAP. 

According to theory 4> should thus mount with t continually, 
though exceedingly slowly, the potential tending to become 
logarithmically infinite. Due to unavoidable leakages, however, 
<J> will practically reach a maximum after some finite time. 

For the thermionic current we find ultimately, by (59) and (60), 




so that the current decreases continually, tending to zero for 
= oo. 

These theoretical results were tested experimentally by 
Richardson. The value of en is determined from the measured 
intensity of the current at the beginning of the experiment and 

that of ----- from the time rate of change of the current, and 
mC 

from these two magnitudes the value of eh/m can be calculated. 

Now, A = 3/2w 2 , where u* is the mean squared velocity of the 
electrons, within as well as outside the metal. Whence, 



m 

It is now assumed that the mean energy of an electron is 
equal to that of a gas molecule, so that |ww a = (ikT, and therefore, 
eh e 



m 

Again, if N be the number of molecules per gram molecule, 

eA -_^L = - E 
m 2kNT 2RT* 

where R is the gas constant per gram molecule, and E (positive) 
the charge of a gram ion of a monovalent electrolyte. The value 
of E is thus known from electrolysis. 

Now, having derived eh/m from his experiments and measured 
the temperature T of the hot metal, Richardson deduced by 
means of the last-written formula the value of the gas constant, 
for a quantity of gas, however, which at the temperature C. 
and the pressure of 76 cm. occupies a volume of 1 cm. 8 . 

Since the value of R per gram molecule (2 grams of hydrogen) 
is 83*2 . 10 6 and since 1 cm. 8 of hydrogen at and 76 cm. weighs 



vi KINETIC THEORY OF SYSTEMS OF ELECTRONS 173 

0-0000898 gram, Richardson's theoretical value of the gas con- 
stant, which will be denoted by R, should be 3730. 

The results of the experiments, in which the wire [platinum 
strip] had temperatures from 1473 to 1813 and was heated 
during 16 up to 35 hours, were as follows : 

A-4-1.10 3 
4-2. 10 3 
3-5 . 10 3 
3-6 . 10 3 
2-9 . 10 3 
3-1 . 10 3 
3-2 . 10 3 
34 . 10 3 

In a later experiment 4-04 . 10 3 was found. 

The satisfactory outcome of these experiments proves that 
the free electrons in a metal have the same mean kinetic energy 
as a gas molecule and that for their velocity distribution Maxwell's 
law is valid. 



30. VELOCITY DISTRIBUTION OF THERMIONS FOR 
DIFFERENT DIRECTIONS 

In the experiments described in Art. 29 it depends only upon 
the velocity component perpendicular to the heated plate whether 
an electron reaches the plate P 2 or not. The agreement with 
Maxwell's law is thus actually proved only for the component . 
In a second series of experiments * Richardson investigated the 
distribution of the velocity components i) and f. The arrange- 
ment of these experiments was such that the charging of the 
upper plate was mainly due to corpuscles which left the hot plate 
in a slanting direction. 

The electrons are here emitted from a long narrow metal 
strip R (Fig. 20) which fills out almost completely a narrow slit 
in the lower plate P lB This strip is placed perpendicularly to the 
plane of the drawing. The plates are very extended. The upper 
plate P 2 has a narrow slit BC parallel to that at R. The electrons 
flying through this slit enter into the metal cylinder E which is 
insulated from the plate. The plate P 2 and the cylinder can be 

* Phil Mag. (6), xvi., 1008, p. 890 ; (6), xviii., 1909, p. 681. 



174 



KINETICAL PROBLEMS 



CHAP. 



E 



alternately connected with an electrometer and both can be 
shifted together in a horizontal direction. Thus it can be deter- 
mined in each position what 
part of the total thermionic 
current passes through the 

p slit BC. 

2 The measurements are 

made while the electric field 
is yet very weak. 

Let us now assume that 
J A Maxwell's law holds for all 

~n ^ the velocity components and 

FIG. 20. that for the emission of 

the electrons there is the 

same distribution over different directions as for gas molecules 
flying through a surface-element. In order to calculate the 
number of particles which, starting from an element dcr placed 
at a point (Fig. 20) of the hot metal strip, pass through the 
slit BC of the upper plate, we construct around a sphere with 
OA =2, the distance of from the upper plate, as radius. Two 
planes laid through and the edges of the slit BC will cut this 
sphere along two great circles (Fig. 21). Let AB=x, <LAOB = i/r, 
and . YOP = 0. The electrons flying across da with a velocity 
u in the direction OP fill 

out a cylinder with da- A x B C 

as base and the nor- 
mal velocity component 
u cos (AP) as height. Their 
number is thus propor- 
tional to cos (AP). If we 
now count up the electrons 
flying through da- with 
different velocities but all 
in the direction OP, the 
result will still be pro- 
portional to cos (AP), and if this direction falls within a narrow 
cone of aperture sin 0d0dyfr, the number of electrons will 
also be proportional to the latter. Since, in the triangle 
APE, cos (AP) =sin cos yfr, that number can be written 
c cos sin 2 d9di/j, 




FIG. 21. 



vi KINETIC THEORY OP SYSTEMS OF ELECTRONS 175 

where c is a constant. Thus the total number of electrons sent 
towards the slit will be 

r* 

c cos <b dJi I sin 2 OdO =irrc cos ifidilf. 
Jo 

Now, tan \lr =#/z, and therefore, ^ = ~. 

cos 2 ^ z 

The required number can therefore be written 
, dx/ z* \ 3 / 2 

2 71 " % 



Richardson's measurements are in good agreement with this 
result. The number of electrons hitting a narrow strip of the 
upper plate attains a maximum for x =0, while for x = o> it tends 
to zero. 

It is thus proved that for the electrons emitted by a hot metal 
the velocity distribution is the same for all directions ; otherwise 
c would not be constant. 

31. THE WORK REQUIRED TO DRIVE AN ELECTRON OUT 
OF THE METAL 

Richardson * undertook also some further investigations with 
the object of measuring the work required to drive an electron 
out of the metal plate. For this purpose he investigated how 
the emission of the electrons depends on temperature, having 
assumed in this connection the number of corpuscles per unit 
volume of the metal to be independent of temperature. 

Let us consider a surface-element of unit area placed in the 
second medium near at and parallel to the boundary of 1 and 2. 
In order to determine the number n of corpuscles passing through 
this element per unit time, we introduce again the function / 
(cf. pp. 164 and 165), viz. for the metal 

2/* -hu* 



and for the second medium 

~ 



e 



* Phil. Trans. A, cci., 1903, p. 497. 



176 KINBTICAL PROBLEMS CHAP. 

We consider first the corpuscles whose velocities are contained 
between v and v + dv and directions of motion between 6 and 
6 +d&, where is the angle between the direction of motion and 
the normal of the boundary surface. 

The number of these corpuscles passing per unit time across 
the said surface-element is 

2A -At 

a e"^ e ZTTV^ sin cos 0d0dv. . . ,(61) 

Integrating this expression over and v we find for the total 
number of corpuscles crossing that surface-element, per unit time, 

2 / -/* -n Zh 

sin cos 0d0 v*e cfo-^vle"**'. (62) 

.'0 dn 

If jSi be the number of corpuscles per unit volume of the metal, 
we have 

f2A -Ai> ' _2* /"a 

a e *Ve efo=a e m X/^' . . (63) 

whence 

* 6 ""*^. - - - (64) 

v 



(65) 



Now, A= - , so that 

2h 3 1 



m mu* KL 

We introduce further the quantity ^- -%le 9 which will be the 
potential energy for a unit of negative charge, and we write 

e ' eN E 



where E is the (positive) charge of a gramion of a monovalent 
electrolyte and R the gas constant. 
The formula for n then becomes 



Here u is proportional to 

By means of this formula, n being observed as a function of 
temperature, the difference ^ 2 - ^, which is positive, can be 
determined. 



vi KINETIC THEORY OP SYSTEMS OF ELECTRONS 177 

Richardson found from his measurements i/r 2 - -^ =4-1 volts. 

Investigations by Wilson * and by Deininger,*)" and a second 
determination by Richardson, { gave a somewhat greater value. 
The mean of these determinations, 5-54 volts, agrees well with 
the value 5-5 volts found by Richardson in a later investigation. 

32. THE DENSITY OF ELECTRONS IN THE METAL 

From Richardson's experimental findings we may also derive 
the number S t which is here considered as independent of 
temperature. For this purpose we transform the formula for n 
by making use of the relation 



whence follows 



e 

- SBS v-ar'" as \ EI 
m V eN m V m E 



The expression under the radical is positive, since e stands for 
the negative charge of the electron. Thus we have 



2T R e -- 



From the value of n that of S t can now be calculated, taking 
^2 " ^i = 5'5 volts. Richardson, however, does not give the value 
of n itself, so that this has to be calculated from other magnitudes 
measured by him. The experimental procedure was as described 
on p. 174 (cf. Fig. 20). The slit and the cylinder E were placed 
at A above the middle of the heated strip R. The cylinder E 
picked then up all the electrons hitting an area of the breadth of 
the slit BC and of the length of 1 cm. The upper plate, the slit, 
and the cylinder could now be shifted in a direction perpendicular 
to the slit BC and thus also perpendicular to the heated 
strip R. In each of the successive positions electrons were 
collected by the cylinder during 30 seconds. The electrons which 
hit the cylinder during a displacement over the full breadth of 
the plate are those which would reach a strip /9, 1 cm. broad, 

* Phil. Trans. A, ccii., 1903, p. 243. 
f Ann. der Phys., xxv., 1908, p. 285. 
J PkSL Trans. A, ccvii., 1906, p. 1. 
Phil. Mag. (6), xx., 1910, p. 205. 
VOL. I N 



178 KINETICAL PROBLEMS CHAP. 

of the upper plate within 30 seconds; the hot strip R is 
perpendicular to the strip ft. 

From Richardson's measurements the charge carried over, 
per unit of time, from R to /? is found to be 1-63 . 10~ 9 coulombs. 
And since the charge of an electron amounts to 1 -5 . 10 ~ 19 coulombs, 
this gives for the number of electrons emitted by R per second 
and hitting & 10 10 . In order to calculate from this number the 
number n of electrons emitted by the metal strip per 1 cm. 2 and 
per second, we note that the number of electrons flying from R 
towards j3 is equal to that sent out by a rectangle of area 
R$ cm. 2 , if R and ft be the breadth in cms. of the strips denoted 
by these letters. In Richardson's experiments, as mentioned 
before, /3 was equal to 1 cm., while the breadth of R amounted 
in some experiments to 0*02 cm., and in some others to 0*04 cm. 
Thus, taking R =0-03 cm., we find n = 10 10 /0-03 = 33 . 10 10 . 

Again, 

2 -^=5.5 . 10 8 electromagnetic units, 



fi=83-2.10 6 
-e/w = l-77.10 7 

T = 1050 + 273 = 1323. 

This gives for the number of electrons per unit volume of the 
metal S l = 10 25 ' 7 . 

The result is not very satisfactory, the number found being 

E 

much too large. This is due to the high value of 

which we will denote by W. In fact, we have found 



Now, in order that the electrons might escape freely we should 
have ^2 = ^, and therefore, 

n= const. tiSj. 

Owing to the factor e~ iy the number of electrons escaping per 
unit area and unit time will be much smaller than what by the 
last formula would correspond to the number (SJ per unit 
volume of the metal and will decrease considerably with decreas- 
ing temperature. The latter is in accordance with facts. _ 

W~ol_ 1 48 
- .~e 

[i.e. 10 25 ' 7 ]. 2 u 



vi KINETIC THEORY OF SYSTEMS OF ELECTRONS 179 

Such a large number of free electrons per unit volume, however, 
is hard to accept. This is even much greater than the number 
of platinum atoms per cm. 3 , for which one finds 8 . 10 22 .* More- 
over, if it be granted that the electrons take part in the heat 
motion and each of them has the mean kinetic energy $kT 9 
such a large number of free electrons would give a much too 
high value for the specific heat of the metal. 

For the number of corpuscles in the second medium we find, 
per unit volume, 




or by (62), 

On the other hand, S can be written 



*, . . . (66) 



so that the density of electrons in the metal should be 
times that in the second medium, as we already know. 



33. DIFFERENCE IN POTENTIAL ENERGY BETWEEN THE 
ELECTRONS INSIDE AND OUTSIDE THE METAL ATOMS 

The conclusion of Art. 32 can be avoided by assuming that the 
electrons are partly bound to the atoms and partly free. In addition 
to the force in the boundary layer which hinders the electrons 
from escaping freely from the metal, there is then still another 
force which binds some of the electrons within the metal to the 
atoms. We assume that the potential energy in the metal within 
an atom has a definite value, and we denote by fa the potential 
energy per unit of negative charge for an electron outside, and by 
fa' that for an electron within the atom, with the understanding 
that fa' is smaller than fa. Further, we denote the part of the 
space occupied by the atoms by a, and therefore the remaining 
part by 1 - a. 

* In fact, 1 gram atom of Pt weighs 195 gr. ; 1 cm. 3 of Pt weighs 21-5 gr., 
and thus amounts to & gram atom, while the number of atoms in a gram 
atom can be put at 68 . 10 22 . 

VOL. I N 2 



180 KINETICAL PROBLEMS CHAP. 

For the number of electrons per unit volume of the metal 
we thus find 



2 /S 



. (67) 

where we have assumed that the medium 1 is in equilibrium with 

the medium 2 and that the density varies with the ^'s as indi- 

f 

cated by the expression e^ **"** . Our task is now to represent 
the observed facts by means of this formula. Assuming S 1 
constant, we have to find how n depends upon T, to compare 
this dependence with the experimental results, and thence to 
calculate fa and fa'. With these exponents we should then be 
able to deduce again S l from n. 

Two extreme cases may be distinguished. The first, corre- 
sponding to fa' =fa, has just been considered. In this case the 
electrons were subjected only to forces within the boundary layer. 

The second limiting case is that the electrons are acted upon 
by forces only within the atoms and that there are no boundary 
forces. In this case fa will tend to fa and fa - fa' will be much 
greater than fa - fa, so that, even if a be small, the first term can 
be omitted in a first approximation. This gives for 8 l the same 
formula as in the previous reasoning, but with the factor a, and 
with fa replaced by fa'. From the experiments it follows then 
again that fa -fa' =5-5 volts. This, however, gives now for S l 
a value which is only the small fraction a of that found before, 
as if the volume were a times smaller and as if the electrons 
within it were very strongly bound. By choosing a small enough 
the proper value for S can be obtained. 

One might object to the formula (67) on account of its being 
based upon the assumption of a large number of electrons in 
each atom. That the formula is at any rate correct * can be 
shown by a method of reasoning due to Gibbs.f 

In fact, the number of systems in a canonic assemblage, with 
co-ordinates contained between q l and qi+dq ly q 2 and qt+dq v . . . 
q n *ndq n +dq n ,is 

. . . dq n , ... (68) 



* See, however, the remark at the end of this article, 
f SUmentari/ Principles in Statistical Mechanics, New York, London, 1902. 



vi KINETIC THEORY OP SYSTEMS OP ELECTRONS 181 

where E q stands for the potential energy which corresponds to 
the given conditions and is proportional to the temperature. 
If all mutual action between the parts is disregarded, the potential 
energy of the system is equal to the sum of the potential energies 
of its parts and the expression for the number of systems is split 
into factors, so that (68) assumes the form 



In our case x l9 y l9 z 1 will be the co-ordinates of a corpuscle and E 
its potential energy. 

The probability that a corpuscle is situated at a given place 
is thus independent of the remaining corpuscles. The ratio of 
the number of systems in which the first corpuscle is within the 
part a of the space to the number of systems in which it is outside 
that part of the space is then 

ae"^ /0 :(l-a)e" AVe , . . . (69) 

where E i is the potential energy of the corpuscle within and E u 
that outside the part a of the space. The same expression gives 
for each system the ratio of the number of corpuscles contained 
within the part a of the space to the number of corpuscles lying 
outside, and thus formula (67) is established. 

In this deduction no assumption whatever was made with 
regard to the number of electrons contained within the part a 
of the space, so that it could even be applied to the case in which 
there are fewer electrons than atoms. What was assumed, how- 
ever, and what may seem objectionable, is that statistical 
mechanics can be applied to the electrons within the atoms. 



CHAPTER \H 
VACUUM CONTACT OP PLATES OF DIFFERENT METALS 

34. POTENTIAL DIFFERENCE FOR VACUUM CONTACT 

IN connection with Richardson's investigations the following 
considerations present themselves. 

Suppose we had two metal plates, both at the same constant 
high temperature, and placed in an exhausted space at a certain 
distance from each other (vacuum contact). If the temperature 
is high enough, electrons will be emitted by either plate. If the 
two plates are of different metals, the number of these electrons 
will not be the same for both, and this will produce a potential 
difference of a determined equilibrium value, which must be equal 
to the potential difference between the metals when placed in 
direct contact. For, if such were not the case, the two plates 
could be connected by a wire and, although everything is kept 
at the same constant temperature, we should have an electric 
current which would clash with the second law of thermo- 
dynamics. 

Let the plates be placed horizontally and let the lower plate 
A have a higher potential than the upper, B, the potential 
difference being <J>. 

By (58), the number of electrons leaving, per second, the 
lower and reaching the upper plate at a potential difference- <I> 
(which diminishes the velocity of the electrons and thus prevents 
some of them from reaching the upper plate) is 



we 



* Cf. p. 176, (65) and (65'). Tn (58) * was the potential of the upper plate 
with respect to the lower one. 



182 



CHAP, vii VACUUM CONTACT OF PLATES 183 

where n is the total number of electrons emitted, per second, by 
the lower plate. In view of (66), n can be written 



where ty a (the previous fa - fa) is the difference in potential 
energy per unit of negative charge within and without the metal 
A, and S a the number of electrons per unit volume of this metal. 
This gives for the number of electrons flying, per second, 
from A to B 



1 W 

\J Ttm 



while the number of those flying, per second, from B to A is 



i I IkT 

% I - 

\J Trm 



the factor e R f ' being here left out, since the electric field does 
not counteract the motion of these electrons, so that all of them 
reach the plate A . 

In the case of equilibrium both numbers must be equal, 
hence the equation 



S 



This gives f er <I> 



a 



The ratio S h /S a is thus in the present case also given by an 
exponential law in which appears the potential energy, split into 
an electrostatic part and one which depends on the difference in 
the attraction between the metals and the electrons. 

The same law for the ratio S b /S a holds also in the case of a 
direct contact between the plates A and B, and since S a and S b 
have fixed values, the potential difference in the latter case is 
the same as for plates separated by a vacuum. 

We can introduce here the same modifications as in the 
preceding investigation by dividing the electrons within the metal 
into two kinds, the intra-atomic and the extra-atomic ones. 

Let the potential energy of an electron in vacuo be by -ep 



184 KINETICAL PROBLEMS CHAP. 

greater than the potential energy of an electron in the metal 
outside the atoms, and by -ep' greater than that of an electron 
within a metal atom. Since e is always the negative value of 
the charge of an electron, both of these expressions are positive. 
Let, further, E/RT=/j,. 
Then, by formula (67), 




For the case of the two plates A and J5 under consideration 
we find, in view of the potential difference <J> (the plate A having 
the higher potential), 



the suffixes 1 and 2 being attached to all magnitudes belonging 
to the plates A and B respectively. 

In the case of a direct contact between the metals the same 
formula will hold for the equilibrium value of the potential 
difference. 

35. RESISTANCE AT A VACUUM JUNCTION 

Let us now consider a thermo-electric element with two 

vacuum junctions which 
are kept at the tempera- 
tures T and T (Fig. 22). 
^ This will produce , a 

current, the electromotive 
force being the resultant 
FIG. 22. of two potential differ- 

ences at the junctions 

while the element is open, and of possible potential differences 
in the metals due to the temperature difference. 

Now, we may ask whether this current has to overcome also 
a resistance at the vacuum junctions. To answer this question 



vn VACUUM CONTACT OF PLATES 185 

we assume that the potential difference <J> between the plates 
differs very little from the equilibrium potential difference <fr . 

From the preceding considerations it follows that the number 
of electrons passing from the plate 1 (of higher potential) to the 
plate 2 is n^"^. The number moving in the opposite direction 
is n 2 , so that the electric current (reckoned positive from 1 to 2) 
will be 

t= eft^e - ***-!!,). 
Again, 



so that 

t 
If fji (4> - <& ) is small, we have approximately 



so that there is, in a certain sense, a resistance at the vacuum 
junctions which is equal to 



__ 1 



This is positive, since e is negative. 



36. PELTIER EFFECT FOR A VACUUM CONTACT 

To close this subject, let us still consider the question of the 
Peltier effect for the case of such a vacuum contact. If the 
effect were here the same as in the case of metal plates in direct 
contact, an electric current should give at the place of junction 
(here vacuum) a heat generation when sent around in one, and a 
cooling when sent around in the opposite sense. 

Suppose that within the enclosure containing the plates 
(Fig. 23) everything is kept at a constant temperature T by means 
of a heat reservoir. We have then to calculate how much heat 
must be supplied or absorbed by the latter when an electric 
current passes through the system. Let the current flow from 
the plate 1 to the plate 2. Then the negative electrons move 
from 2 to 1. The heat taken up by the reservoir during a given 
time will be equal to the energy of the electrons which during 
this time enter into the system at P less tke energy of those 



186 KINETICAL PROBLEMS CHAP. 

which, during the same time, leave the system at Q, both the 
kinetic and the potential energy being taken into account. 

Since the temperature is the same at P and Q, there will be 
no difference in kinetic energy. Let the potential of the plate 1 
be higher than that of the plate 2, and let their potential difference 
<$> differ but little from its equilibrium value. Consider a quantity 
of electricity - 1 streaming from P towards 
Q. Owing to the potential difference of 
the plates, this negative charge has at 
the entrance P into the system a greater 
potential energy than at the exit Q, the 
difference being just 4>. In addition to 
this we have also to take account of the 
Fia. 23. ^ potential energy of the electrons due to 
the attraction by the metal atoms. The 
state being stationary, the number of electrons contained 
within the atoms will remain unaltered. Thus we can imagine 
that the electrons [involved in the current] pass between the 
atoms. If the potential energy of an electron in the vacuum is 
higher by - ep l than that of an electron in the plate 1 outside 
the atoms, and by -ep% higher than that of an electron in the 
plate 2 outside the atoms, then the negative unit of charge loses 
in passing from P to Q the amount of energy p l -p 2 . [This is 
to be added to <fr.] 

Thus the total amount of energy lost by the electrons and, 
therefore, gained in the form of heat by the reservoir is, for each 
unit of positive charge passing from 1 to 2, or of negative charge 
passing from 2 to 1, 




We find, therefore, for the Peltier effect TT, 



37. RICHARDSON'S MEASUREMENTS 

The heat generated in such a system has been measured by 
Richardson.* 

In a vacuum was placed a cold plate P , kept at a temperature 
T , and opposite it a hot plate P, whose constant temperature T 

**Phil. Mag. (6), xx., 1910, p. 173. 



vn VACUUM CONTACT OF PLATES 187 

was so high as to make it emit electrons. In order that all these 
may reach the plate P , the potential of the latter was kept an 
amount <P above that of P. 

Under these conditions Richardson measured the heat 
generated on the plate P . 

We consider the plate P together with the vacuum next to 
P and a part of the wire attached to P (and consisting of the 
same metal as P ) as a system whose gain in energy has to be 
determined. This will be due, in the first place, to the kinetic 
energy of the electrons entering the system. In order to deter- 
mine this energy we require the mean squared velocity of the 
electrons flying into the vacuum. Now, the number of electrons 
emitted per unit time and unit area by the plate P, with velocities 
contained between v and v + dv, is * 



where a is a constant. Consequently, the mean squared velocity 
with which the electrons enter the vacuum, 

rfdv /73 o A 

Trallr z 4 , 

__ . [_ . 1 1 A T 

.'o 7rfle 

where * a is the mean squared velocity of the electrons in the 
metal. The mean velocity of the electrons is thus greater at 
their arrival in the vacuum than within the metal, which is to 
be explained by the circumstance that only the electrons endowed 
with the greatest velocities leave the metal. J 

By what precedes, in conjunction with mfih^kT, the kinetic 
energy entering the system per unit time can be written 2nkT, 
where n is the number of electrons passing per unit time from 
P to P . As against this, the kinetic energy of the electrons 
escaping through the wire amounts to | -nkT Q . 

Now for the potential energy of the electrons. In the first 
place, owing to the potential difference <J> of the plates the energy 
of the electrons entering the system exceeds by -en that of 

* Cf. p. 170, (61), after integration over 0. -f Cf. p. 137. 

J This is disregarded by Richardson. The final result, however, remains 
unaffected, since the difference in kinetic energy between the electrons moving 
in and out is, after all, negligible when compared with the difference in potential 
energy. 



188 KINETICAL PROBLEMS CHAP, vn 

the electrons leaving the system (e being always negative). In 
the second place, owing to the attraction by the metal atoms, 
bhe potential energy of an electron will be greater outside than 
within the system. Denoting, therefore, this excess by -ety we 
shall have for the heat generated on the plate P 

W - n(2kT - 4 kT - eO - &!*). 



[n Richardson's experiments e^+e-^ was so much greater than 
that we may as well write 



The plate P was inserted in one of the branches of a Wheat- 
stone bridge, so that the heat generated could be measured 
through the change of the resistance. Richardson made use of 
the circumstance that IF is a linear function of <E>. Taking his 
measurements at different values of 4> and determining the ratio 
of the W's, he was able to calculate ty. 

This quantity appeared to be different according to the manner 
in which the piece of platinum was treated before the experiment, 
and depends probably on the presence or absence of occluded 
gases. The value found for ifr by Richardson was again 4 to 5 
volts, and thus agreed with the value found by previous measure- 
ments (cf. Art. 31). 

There remains, however, yet one great difficulty. We saw 
(Art. 33) that the potential difference of 5-5 volts which was 
obtained in Richardson's experiments described in Art. 31 had 
to be explained mainly by the difference in potential energy of 
the electrons in the vacuum and of the intra-atomic ones, viz. 
by assuming that the electrons are strongly bound to the atoms. 
Now, in connection with the last experiments there is no question 
of electrons in the atoms, so that the potential difference of 4 to 
5 volts must here be explained by the difference in potential 
energy of the electrons contained in the metal but between tHe 
atoms and the electrons in the vacuum. Thus the agreement of 
the results of the two sets of measurements is rather unintelligible, 
and it must be admitted that not all is clear in these investigations. 



CHAPTER VIII 
PROBLEMS IN WHICH THE MOTION OP ELECTRONS PLAYS A PART 

38. NICHOLS* EXPERIMENT 

THE presence of free electrons in a metal suggested to Nichols * 
the following experiment. He set a metal disc into rapid spinning 
motion about its axis, expecting the electrons to be driven by 
the centrifugal force towards the rim and thus to produce a 
potential difference between the rim and the centre of the disc. 
Let -e be the charge and m the mass of an electron, &> the 
angular velocity of the disc, and 3> the potential at a point of the 
disc at a distance r from its centre. Then the condition of 
equilibrium is 

2 d& 

ma) z r=* -e J-, 
ar 

whence, by integration, 

i _ 

O = - ~ oj 2 / 2 + const., 

or, if <!>! be the potential at the centre and <E> 2 that at the rim 
(72 being the radius of the disc), 



The question is whether this potential difference can be detected. 
If we take 



sec.- 1 , 
& 
= l-8 . 10 7 (electromagnetic units), 



then 

. O x - O a - 1-1 electromagnetic units 1-1 . 10~ 8 volt. 

* Phys. Zeitschrift, vii., 1906, p. 640. 
189 



190 KINETICAL PROBLEMS CHAP. 

Nichols spent much time in trying to detect this potential 
difference, but his apparatus seems not to have been sensitive 
enough. Owing to the rotation of the disc, the measurement of 
the potential difference was connected with great difficulties ; 
sliding contacts had to be applied. He arrived at the conclusion 
that a potential difference 1000 times as great, such as should 
manifest itself if the positive ions were free to move, could be 
detected by his apparatus. The absence of the effect was thus a 
proof that the positive ions are bound.* 



39. KINETIC ENERGY OF THE CONDUCTION-CURRENT ELECTRONS 

The following question suggests itself in connection with the 
preceding considerations. Can the magnetic energy &Li 2 of an 
electric current, which is often referred to as the [electro-] 
kinetic energy, be interpreted as the total kinetic energy of the 
electrons, S(Jwv 2 ) ? The answer is decidedly in the negative. 
For \Li* depends on the self-induction and, therefore, on the 
shape of the wire, and is in common cases much greater than 



The kinetic energy of an electron, whether entirely or partly of 
electromagnetic nature, is at any rate localised in the immediate 
neighbourhood of the electron. The total kinetic energy of all 
electrons is found by addition, since the spaces in which this 
energy is located do not overlap. This gives ^(^niv 2 ). On the 
other hand, the quantity \L& represents the total magnetic 
energy to which every electron contributes through its weak 
magnetic field which extends to a considerable distance. Tn the 
case of an electric current, however, the magnetic fields of different 
electrons are equally directed and will, surely, overlap. Since 
the magnetic energy is a quadratic function of the magnetic 
force, the total magnetic energy will not be obtained by simply. 
adding up the energies due to the separate electrons. Thus, for 
instance, if there are N electrons and if each of them produces 
at a given point of space the same magnetic force // of exactly the 
same direction, the resultant magnetic force will be NH, and the 

* [This experiment has since been carried out successfully, in a modified 
form and with more refined means, by II. C Tolman and others. For its 
recent history and the results obtained see Physical Review, vol. viii., 1910, 
p. 97 and p. 753 ; vol. ix., 1917, p. 164 ; vol. xxi., 1923, p. 525 ; vol. xxii., 
1923, p. 207.] 



vni PROBLEMS IN MOTION OP ELECTRONS 191 

magnetic energy will be proportional to 2V 2 , while the quantity 
S(^mv 2 ) is proportional to N itself. 

Since the number of electrons involved in an electric current 
is very large, it will be seen that \Lfi may be much greater than 
S(4it*). 

For the sake of illustration let us consider a simple case in 
which the conductor consists of a cylindrical wire sheathed by 
a co-axial tube. Let the current i flow upwards in the wire and 
downwards in the sheath. The magnetic force at a distance r 
from the axis is 

H- 



'2<rrcr 

where c is a constant, the propagation velocity of light. If a t 
and fl 2 t> e the radii of the wire and the tube, the magnetic energy 
in the space between them is, per unit length, 

/.log* (70) 

47TC 2 n a l x ' 

On the other hand, if N t and N 2 be the numbers of the electrons, 
per unit length, in the wire and the tube, respectively, and v l9 v 2 
their velocities, the kinetic energy of the electrons will be 

im(#iV+W) (71) 

Let us now compare the expressions (70) and (71). 

We have i = Jf l ev l = N^BV 2 an( i> using the electromagnetic mass, 



6 

m = n n - 9 (if R be the radius of an electron). Thus (71) becomes 

_. i i._ a ( 1 + l). 



Now, if a 2 l<*i is a moderate number, (71) will be much smaller 
than (70), provided that 



is a very small fraction. 

For that purpose RN l and RN 2 must be very large, that is 
to say, the number of free electrons in the conductor, contained 
between two parallel planes at a distance equal to the radius of 
the electron, must be very great. 



192 KINETICAL PROBLEMS CHAP, vnx 

Take 5 = 1-5 . 10 ~ 18 cm. and consider a copper wire of 1 cm. 2 
cross-section. Then the number of centres of metal atoms 
contained between the said planes will be 14 . 10 9 . 

This number is so great that the number of free electrons can 
satisfy the requirement of making RN very large, and yet be 
small compared with the number of atoms, the latter condi- 
tion being indispensable in order to avoid difficulties with regard 
to specific heat. 

Only for extremely thin wires (of a diameter of the order 
of a wave-length of light) would the value of \Li* be comparable 
with that of 



SOURCES AND REFERENCES 



CHAFTEB I 

EL Helmholtz und G. v. Piotrowski, Vber Reibung tropfbarer Fliissigkeiten, 

Wiener Sitzungsber. xl., Abt. I., 1860, p. 607. 
\. Einstein, Zur Theorie der Brown* schen Bewegung, Ann. der Phys. xix., 

1906, p. 371. 

CHAPTER II 

J. C. Maxwell, On the Viscosity or Internal Friction of Air and other Gases, 

Phil. Trans, clvi., 1866, p. 249. 
&. Kundt und K. Warburg, Uber Reibung und Warmeleitung verdiinnter Oase, 

Ann. Phys. u. Chemie, civ., 1875, pp. 337 and 525. 

CHAPTER III 

M. Knudsen, Die Gesetze der Molekularstromung und der inneren Reibungs- 

stromung der Gase durch Rohren, Ann. der Phys. xxviii., 1909, p. 75. 
Die Molekularstromung der Gase durch Offnungen und die Effusion, Ann. 

der Phys. xxviii., 1909, p. 999. 
Eine Revision der Gleichgewichtsbedingung der Gase. Thermische Mole- 

kularstromung, Ann. der Phys. xxxi., 1910, p. 205. 
Thermischer Molekulardruck der Gase in Rdhren und porosen Kdrpern, 

Ann. der Phys. xxxi., 1910, p. 633. 

Kin absolutes Manometer, ibid, xxxii., 1910, p. 809. 

Eine Methode zur Bestimmung des Molekulargewichts sehr kleiner Gas- 

oder Dampfmengen, ibid, xliv., 1914, p. 525. 
J. E. Shrader and R. G. Sherwood, Production and Measurement of High Vacua, 

Phys. Rev. (2), xii., 1918, p. 70. 
M. Knudsen, Die molekulare Warmeleitung der Gase und der Akkomodations- 

koeffizient, Ann. der Phys. xxxiv., 1911, p. 593. 
M. v. Smoluchowski, Ober Warmeleitung in verdlinnten Gasen, Wied. Ann. 

Ixiv., 1898, p. 101. 

Vber den Temperatur sprung bei Warmeleitung in Gasen, Wiener Sitzungs- 
ber. cvii. Abt. Ha, 1898, p. 304, and cviii. Abt. Ho, 1899, p. 5. 
P. Lasareff, Vber den Temperatur sprung an der Grenze zwischen Metall und 

Gas, Ann. der Phys. xxxvii., 1912, p. 233. 

CHAPTER IV 

G. L. Lesage, Loi qui comprend touted lea attractions et repulsions, Journal des 
Savants, 1764. 

193 



194 KINETICAL PROBLEMS 

A. Einstein und L. Hopf, Statistische Untersuchung der Bewegung eines Resonators 

in einem Strahlungsfeld, Ann. der Phys. xxxiii., 1910, p. 1105. 
H. A. Lorentz, Rapport sur ^application au rayonnement du theoreme de 

requipartition de Venergie. Cas particulier d'un electron libre, Rapp. Reunion 

de Bruxelles, 1911, p. 37. 
A. D. Fokker, Snr les mouvements browniens dans le champ du rayonnement noir, 

Arch, neerl. (Ilia), iv., 1918, p. 379. 

CHAPTER V 

G. Stokes, Mathematical and Physical Papers, Cambridge, 1880, i. p. 100. 

Rayleigh, The Theory of Sound, London, 1896, ii. p. 312. 

G. Kirchhoff, Vber den Einfluss der Wdrmeleitung in einem Oase auf die 

flchallbewegung, Pogg. Ann. cxxxiv., 1808, p. 177 ; Ges. Abh., Leipzig, 

1882, p. 540. 
H. Kayscr, Bestimmung der specifischen Warwe fur Lvft bei constantem Druck 

und constantem Volumen durch Schallgeschwindigkeit, Wied. Ann. ii., 1877, 

p. 218. 
P. Lebedew, Die Grenzwerte der kiirzesten akustischen Wellen, Ann. der Phys. 

xxxv., 1911, p. 171. 
N. Neklepajev, Ober die Absorption kvrzer atustischer Wellen in der Lufl, Ann. 

der Phys. xxxv., 1911, p. 175. 
J. H. Jeans, The Dynamical Theory of Gases, Cambridge, 1910, second edition, 

p. 374. 
H. A. I^orentz, Les equations du mouivment des gaz et la propagation du son 

suirani la theorie cinetique des gaz, Arch, neerl. xvi., 1881, p. 1. 



CHAPTER VI 

0. W. Richardson and F. C. Brown, The Kinetic Energy of the Negative Electrons 

emitted by Hot Bodies, Phil. Mag. (($), xvi., 1908, p. 353. 
0. W. Richardson, The Kinetic Energy of Ions emitted by Hot Bodies, Phil. Mag. 

(6), xvi., 1908, p. 890, and xviii., 1909, p. 081. 
The Electrical Conductivity imparted to a Vacuum by Hot Conductors, Phil. 

Trans. A, cci., 1903, p. 497. 
The Emission of Electricity from Hot Bodies, London, 1910. 

CHAPTER VII 

0. W. Richardson and H. L. Cooke, The Heat developed during the Absorption 
of Electrons by Platinum, Phil. Mag. ((5), xx., 1910, p. 173 

CHAPTER VIII 

E. F. Nichols, Die Moglichkeit finer durch zentrifugale Beschleunigung erzeugten- 

elektromotorischen Kraft, Phys. Zeitschr. vii., 1900, p. 040. 
A. Schuster, On Electric Inertia and the Inertia of Electric Convection, Phil. Mag. 

(0), L, 1901, p. 227. 
H. A. Lorentz, The Theory of Electrons, Leipzig, 1909, p. 47. 

Further developments of Richardson's thermionic theory were made, among 
others, by L Langmuir, W. Schottky, J. A. Fleming : 

1. Langmuir, The Effect of Space Charge and Residual Oases on Thermionic 
Currents in High Vacuum, Phys. Rev. (2), ii., 1913, p. 450. 



SOUKCES AND REFERENCES 195 

Thermionenstrdme im hohen Vakuum. I. Wirkung der Raumladung. 

II. Die Elektronenemission seitens des Wolframs und die Wirkung von Gas- 

resten, Phys. Zeitschr. xv., 1914, pp. 348, 516. 
W. Schottky, Die Wirkung der Raumladung auf Thermionenstrdme im hohen 

Vakuum, Phys. Zeitschr. xv., 1914, p. 526. 
Ober den Einfluss von Potentialschwellen auf den Stromiibergang zwischen 

einem Gluhdraht und einem koaxialen Zylinder, Phys. Zeitschr. xv., 1914, 

p. 624. 
Vber Raumladungswirkungen bei Strdmen positiver lonen im hohen 

Vakuum, ibidem, p. 656. 
Vber den Einfluss von Strukturwirkungen, besonders der Thomson* schen 

Bildkraft, auf die Elektronenemission der Metalle, ibidem, p. 872. 
J. A. Fleming, The Thermionic Valve and its Developments in Radiotelegraphy 

and Telephony, London, 1919. 



Printed in Great Britain by R. & R. CI.ARK, LIMITED, Edinburgh.