G. F. Firzazratp—On the Mechanical Theory of Crookes's Force. 63 
and the pressures deduced from this formula are very much smaller than those 
observed, so that it seems certain that the hypothetical distribution Clausius 
assumed is not at all adequate to represent the actual one. The pressures obtained by 
this formula are so insignificant that it is not worth while giving the details of the 
method by which it is deduced. That Clausius hypothesis is by no means 
adequate, can also be seen by the consideration that itis only after the Clausian 
laws for the conduction of heat have ceased to apply owing to the rarefaction of 
the gas that Crookes’s force becomes remarkable as well as by considering what 
the distribution tends towards in this case, when the number of molecules is small 
compared with the distance between the heater and cooler, as has been done by 
Mr Stoney in his paper read before this Society at its last meeting. He shows, as 
is also pretty evident, that the distribution tends towards one which could be 
represented by two unopposing streams of molecules moving one towards the heater 
and the other towards the cooler. With such a distribution the laws of conduction 
of heat would of course differ somewhat from those deduced from Clausius’ 
distribution. 
I shall now calculate the result upon an arbitrarily assumed distribution, which, 
however, probably represents the actual one more nearly than Clausius’. I shall 
assume that the distribution of velocities can be represented by the formula— 
v=0,(1+a cos 0+ sin @ sin $+ y sin 6 cos ¢ 
+a cos 0+ 6 sin’ @ sin? ¢ +¢ sin 7@-cos *o + 2f-sin 70 sin ¢ cos ¢ 
+ 2g sin 8- cos 0: cos $+ 2h sin @- cos @- sin ¢ 
where cos 0=p 
This is equivalent to saying that it 1s represented very nearly by the. radii drawn 
to the surface of a slightly elliptical ellipsoid from a point near its centre. I shall 
assume that «By abe fgh are all quantities whose squares and products may be 
neglected. For the number of molecules moving in the given direction 9, ¢ I 
shall assume that it varies inversely as the velocity of the molecules moving 
in that direction so that nv=Nv, This evidently satisfies the condition A,~=o. 
By these assumptions we obtain approximately nv’=Nv,'v and nv’=Nv,'v? and hence 
(OE Na One | 
me—=Nv2 2 +a p?+b(1 — p?) sin? $ + ¢(1 —p2) + cos? o L 
+2fV1—p? sin @ cos 6+ 29 wW1— p< cos 6 + 2h pW 1 — p? «sin @) | 
or turning it into the form of a series of spherical harmonics 
eae 1 } 
1+ (2 +b+0¢)+ (a—96 +¢)(p? — 4) +5(¢—6)(1 — p?) cos 26 
nme—N v,? == ——; —— . 
+2fV1—p?-sin @ cos + 2g pW 1—p?- cos $+ 2f pV 1 — p? «sin o 
{ +apu+ BV 1— pe? * sin o+yV 1— * COS @ 
from which we see that 
1 1 
b,=a—5(b +c) bs=5(e—6) 
b,=27 b,=2g b6,=2h 
