349-351] Collisions with Law of Inverse Fifth Power 293 



Squaring the system of three equations (698), (699) and (700), and 

 adding, 



u 2 + v 2 + w 2 = 2 (w + a) 2 - 22 (u + a) a' cos (e - co,} + 2a' 2 cos 2 (e - Wj), 

 so that 



u (u 2 + v 2 + W-) = (u + a) 2 (u + a) 2 -2(u + a) 2a' cos (e - co,) 



+ (u + a) 2a' 2 cos 2 (e - eo x ) 

 + a' cos e2 (u> + a) 3 2a' cos e2a' 2 cos (e G^) 

 + a' cos e2a' 2 cos 2 (e &>i). 



Hence, on integration with respect to e, 

 1 /_ 



27T./0 



= u^u 2 + (u + a) (2 (u + a) 2 -f 2a' 2 ) 2a'2a' 2 cos &) : (701). 



To simplify this, we notice that 



o/2a' cos &>! = sin 2 0' { F 2 (w' - u) 2 + (u f u) (v' v) + (u' u) (w' w)}, 

 Z(u + a) 2 = 2w 2 + 22w (u - u) cos 2 0' + F 2 cos 4 \& ', 



2a /2 = sin 2 0'2 ( F 2 - (uf - u) 2 ) = F 2 sin 2 \& cos 2 0' ; 

 so that we also have 



2 (u + a) 2 + I2a' 2 = 2w 2 + 2 (u 2 - u 2 ) cos 2 0'. 



It is clear therefore that the right-hand side of equation (701) can be 

 expressed as the sum of two terms multiplied by cos 2 0'/2 and sin 2 0' 

 respectively. Simplified as far as possible, we find for this expression the 

 value 



(i/'^ti' 2 Sw 2 ^ on<5 2 10' 



^ Ct/ '' '' '' J \j\J?> o i/ 



_i_ 1 cin 2 0' F^/ ^9^7/' 2 5 1 ?/ 2 Vi/;/^ 4- ' l^Stfi Vi/' 2 _ Ti/j/'M 



j^ ^ olll I/ I cfc ^ j. ff' ^c(/ Kit } ^r Lb ^ .'(/ _ ff ^w'f '' ^J. 



After integrating with respect to a, and averaging over all values of the 

 velocities, it is obvious that the first line vanishes, while from the second we 

 obtain, by the use of equation (691), 



"Pj" 2 



J J IU{U 



e=0 a=0 



I I sin 2 0'ada 



a = 



+ u 



- 2u (u 2 + v 2 + w 2 ) - 2 (u u? + v uv + w uw)\ 



(702). 



