G. F. Frrzarratp—On the Mechanical Theory of Crookes’s Force. 61 
will occur as squares in the spherical harmonic of the 2nd order in nv’ so that b,, by 
will contain ¢, ¢, andc; in the second degree, i.e. will contain terms varying as the 
squares ofthe quantities of heat passing. Itis also to be observed that terms occur- 
ring in the spherical harmonics of the 2nd order can never come into those of the 1st, 
except as products with terms belonging to spherical harmonics of the 3rd order so 
that a hypothetical distribution which gave correct values for the quantities of heat 
passing might very well be quite inadequate as a means of calculating the difference 
of pressure in different directions. This remark is of importance when we come to 
consider the results of Clausius’ hypothesis and was suggested to me by Mr, Stoney in 
conversation. | 
As an example of what I am insisting upon,.we may take two opposite extreme 
cases. First, the case of B, vanishing, and secondly the case of C, doing so. In the 
first case there would be a distribution of velocities and numbers such that though 
heat would be conducted across the layer nevertheless there would be no resultant 
inequality of stress, while in the 2nd case though no heat would be conducted yet 
there would be inequality of stresses. It seems, however, certain that neither 
of these extreme cases can exist as a permanent distribution in gases. Before 
calculating the values of these quantities upon particular hypothetical distribu- 
tions, it may be well to see what they are in the simple case of two parallel 
planes each at a uniform temperature. 
In this case itis evident fromsymmetry that if we take X normal to the planes 
we must have all our equations independent of ¢ as the effect 1s evidently 
symmetrical with regard to X. Then we get 
6, = 6, = b, = b, =o =c4,=— 6; 
and there are no tangential forces while all the heat is transferred in the direction 
X and our pressures become— 
1 4 
P,,=3MNv,” (B, + Td b,) 
2 
1 
a P,.= 5MNv,?(B, = 15 
b,) 
while the heat transferred 1s— 
Q.=aMNos @, 
The excess of pressure in X over that in the normal directions is 
2 
Pay = Py=T5MNe,” (Ys 
and this has been called Crookes’s force. 
That it depends wholly upon 0,, can be seen by the following simple method 
mentioned to me by Mr. Stoney : 
Our expressions for P,,, and P,, are 
Pi Menv yu? P= M3nv2(1 — p22) sin? » 
