111-113] System of Loaded Spheres 99 



Let the result of averaging in this manner be denoted by {q}. 

 When we require to average any quantity p over all possible positions of 

 the axes of the first sphere, all positions being regarded as equally probable, we 



shall denote the result by Ipdw, and the same quantity averaged over all 



positions of the axes of the second sphere will be denoted by \pdw. The 



probability of an impact occurring for any given positions of the axes is, 

 however, proportional to F, arid this depends on the positions of the axes. 

 Thus, averaging over all collisions for which the coordinates (239) have given 

 values, we shall have 



q Vdvdv' 



........................... < 240 >- 



Vda>da>' 



r 



1 1 



To calculate {c 2 } from equation (238), we must first evaluate {/j}..., 

 {Ijli} ..., [D], and [AV' 2 cos 2 </>}. Since, however, we are neglecting r 3 , we 

 require {l^ ... only as far as terms in r, and {IJi} ... and the remaining 

 averages only when r = 0. 



From equation (226) 



F 2 = 2 {u' - u + r (^ V - /s'tj/ - ^OTJ + l^}}*, 



so that if we write 



f7 2 = (u - uj + (v' - vf + (w' - w}\ 



so that U is the relative velocity of the centres of gravity, we have, as far as 

 terms in r, 



so 



Now obviously I Ipdcoda)' = 0, when p has any of the values 



li, 1% ..., LI, 1 2 ..., i\L\ ... etc., 

 that [ \Vdvda)' = U as far as terms in r, and hence, from formula (240), 



r r 



and since obviously nlfdwdw = ^, this becomes 



With the help of this and similar other expressions, we have as the 

 averaged value of one term in equation (238) 



,2 ( (V-.'-...)l = ^^2'(.'-> 



72 



