109-111] System of Loaded Spheres 97 



By [q] we shall denote the average value of any quantity q taken over 

 all collisions in which the two spheres are, before impact, in certain specified 

 conditions. 



By squaring and adding equations similar to (224), 



27 7 2 



u- + v 2 + w- = u? + v 2 + w 2 H (\u + LLV + vw) H (230). 



TO v ra 2 



Before averaging this equation, we must calculate 



r/n l~/ "1 



and (\u + uv + vw) . 

 |_ra 2 J \_m ^ 



If V is the relative velocity, 



\OL + /A/3 + vy = V COS >/r ; 

 hence by equation (229), we have as far as r 2 , 



-, = F 2 cos 2 -b - 2r*A F 2 cos 2 xfr. 



TO 2 



riT 



Now [ F 2 cos 2 i/r] = F 2 2 sin ^ cos 3 +4+*=$ F 2 . 



Jo 



Hence f^l = | F 2 - 2r 2 [A F 2 cos 2 -f ]. 



Again, from equation (229), 



- (\u + pv + vw) = S [X 2 ] an + S [>z>] (/9w + TW) 



-r 2 {2[^X 2 ]aM + S[^H(^ + 7 v)} (231). 



111. If we take the direction of the relative velocity for pole, and denote 

 the coordinates of a point on a unit sphere, referred to this pole, by , <&, 

 then the proportion of cases in which the direction (X, /u, v) meets this unit 



sphere within limits dSd<& is - sin cos d d<&. Hence 



7T 



ft /-27T 



= - X 2 sin cos d0 d^> (232). 



T J ^0 



If we take the coordinates of the axis of x to be (6, </>), we have 



X = cos cos 6 sin 8sin0cos(< - </>) (233). 



4 

 Substituting this value for \ in equation (232) and integrating, we 



obtain 



[X 2 ] = cos 2 + i sin 2 



.(234), 



and similar equations give [//. 2 ], [v 2 ]. 

 j. 



