62° G. F. Firzceratp—On the Mechanical Theory of Crookes’s Force. 
so that calling 
— = I dudo 
when I depends upon the distribution of numbers only we can write the pressures 
MN MN 
Pa= qf lwurdude Py qf Ter — p?) sin” pdudg 
We can integrate them with respect to ¢ for we know that Iv” is independent 
of » in the case we are considering— 
1 1 3 
*. Pup MNfIo dp — P,y—=ZMNJTo(1 — p?)dp ©. Per — Py=K=7MN/To%(u" = dda 
So that if Iv’ be expanded in spherical harmonics, K depends only upon the 
spherical harmonic of the 2nd order. Similarly if Iv’ be similarly expanded, it is 
easy to see that 
Qu 5 MNI/ Le pd 
can only depend upon the spherical harmonic of the 1st order in Iv’. 
If now we turn to particular hypotheses as to the character of the distribution 
of velocities and numbers the first that claims our attention is Clausius’s. He 
starts from the assumption that the distribution of velocities among the molecules 
that have just encountered one another in any given layer, may be perfectly represented 
by supposing a small constant velocity in the direction of the transference of heat 
to be superposed upon a uniform distribution. This is the same as supposing . 
that these velocities in various directions may be represented by the radi drawn 
to the surface of a sphere from a point slightly displaced from its centre. It is 
worthy of remark, in connexion with what I mentioned before with reference to 
the way the quantities in the various spherical harmonics are related to one another, 
that supposing the sphere to be an ellipsoid of even great ellipticity would not have 
affected his results, for it is easy to show that the ellipticity of an ellipsoid of 
revolution only enters into the spherical harmonics of the 2nd and higher orders so 
that it would not enter into the equation giving the quantity of heat except when 
multiplied by terms of at least the order of the quantity of heat. Thus even 
though the square of the ellipticity were of the order of the displacement from the 
centre of the point from which the radii representing the velocities are drawn, 
nevertheless that would at most only have introduced terms depending upon the 
product of these two, which would not have materially affected his results. Hence, 
we see that Clausius’s success in calculating the quantity of heat conducted is no 
proof that his hypothesis is by any means a sufficient representation of the actual 
distribution for the purpose of calculating the resultant stresses, and that it is not, 
is proved by calculating what the Crookes’ force would be upon his hypothesis. If 
this be done with the help of the quantities he gives in his note, (see Phil. Mag. vol. 
23, 4th Ser., p. 526) we get— 
