98 The Law of Distribution [CH. v 



If we suppose the axis of y to be (0', <'), then 



fj, = cos cos 0' sin sin 6' cos (^> <'), 



and multiplying this by the respective sides of equation (233) and integrating, 

 we obtain 



I \iLL \ A COS C7 COS t/ ~ ~. - r ~ ..................... i i_->') I, 



4* r 



and similar equations give [/>], [vX]. 



112. Substituting these values in equation (231), 



where D = S[^X']aw + 2 [4/;] (/3w + yv) ............... (236). 



Or, simplifying, 



- (\u + pv + vw) = | (aw + ftv + yw) r 2 D, 



and hence from equation (230), 



{u 2 +tf + uJ 2 ] = u? + v'* + w'*+au + /3v + vw+ J F 2 



- 2r 2 (D + [4 F 2 cos 2 ^]) ...... (237). 



On the right-hand side the upper line 



= \ {u* + v 2 + w* + (it + a) 2 + (v + ) 2 + (w + 7) 2 } 

 = 2tt s + ^S {w' + r (^W - ^Vj' - ^^ + ^i) 2 }, 

 by equation (226), 



If therefore we write 



w 2 + t? + w 2 = c 2 , 



w 2 + i; 2 + w z = c 2 , 



-STj 2 + <GT 2 2 = OT 2 , 



and adopt a similar notation in accented symbols, equation (237) may be 



written 







[c 2 ] = i (c 2 + c' 2 ) + r 2tt' (/>/ -...) + I?- 2 (sr 2 + w' 2 ) 



- r 2 2 (i^nr,^ -...)- 2r 2 (Z) + [A F 2 cos 2 -f]) ...... (238). 



113. We must now carry the process of averaging still further, so as to 

 apply to all possible positions of the axes of the colliding spheres ; that is to 

 say, we must average over all collisions determined by given values of the 

 ten velocities 



11, v, w, u, v', w', wj, Br 8 , ID-,', tsT 2 ' ............... (239). 



