10 
MR. S. H. BURBURY ON THE APPLICATION 
As the number of spheres included in our group diminishes, the coefficients a 
diminish, and, since every b is negative, the b ’s increase in absolute magnitude.. 
On the other hand, as n increases the a’s increase, and every b 2 , if changed at all, 
diminishes. Therefore, as u increases the function Q „ tends to a limiting form. 
But that limiting form must be wT if k = 0, because in that case we know the 
law to be e _7,T . We must then have ultimately Q „ = n (T - j - k0), where 9 is a 
quadratic function of the velocities. And we may now assume that for a sufficiently 
great number of spheres comprised in a group, throughout which k is sensibly 
constant, the law is expressed by the function Ce“ 7lM (T + * T d 5 in which C = -Jr=y/D, 
D being the determinant of the coefficients of the quadratic function T -f- kTV. 
We have thus obtained certain conditions which the coefficients a x , b 12 , &c., must 
satisfy. Another condition is that the assumed law of distribution of velocities 
expressed by the function 
7; (ajii, 2 + + &c.) 
shall not be disturbed by a collision taking place between any two of the n spheres, 
which collision changes the velocities, but not the positions, of the two spheres in 
question. To find the values of cq, b 1Zi &c., to satisfy this condition presents consider¬ 
able difficulty. It is not, however, at present necessary to solve the problem in that 
form, as will be seen later. For we have only to consider the positions of the n spheres 
as unknown, and we obtain a solution sufficient for our purpose. If, namely, it be 
given that there are at any instant n spheres within a spherical space S, but nothing 
is known of their positions within S, we have only to assume that the chance of their 
having at that instant velocities u 1 . . . u x 4- du x , &c., is 
e~ hQi du 1 . . . dw tll 
with 
Q = at ( u 2 + v 2 + iv 2 ) + btt (iiu + vv' -f ivid) 
containing only one coefficient a and one b, and we shall find that all necessary 
conditions are satisfied, including the condition that the assumed distribution shall be 
unaffected by collisions. 
For let aq, y x , %, x z , y. 2 , z 2 be the component velocities of two spheres before collision. 
A collision between the two converts these components into x\, y\, z\> x\, y\ 2 , z\ in 
the following manner. Let g, v be the direction cosines of the line of centres at 
collision. 
The velocities of the two spheres resolved in the line of centres are, before 
collision, 
\x> + gih + vz x and \x 2 + /xy 3 -j- vz 2 
respectively. And we have 
