12 
MR, S. H. BURBURY ON THE APPLICATION 
This, it will be remembered, is on the assumption that all the coefficients a are the 
same, and all the coefficients b are the same, so far as they have to do with the 
colliding spheres. 
18. Now 
Yi n (T + «T,) 
( , n — 1 \ m , 2 k k , /, , n — 1 \ , . 
—- 1 ~b - K ~n~ — UiU., — WiMo -f- 1 -f- K + &c. 
\ n ) 2 n 1 z n 1 ° \ n ) 2 
— alt (hr + v 2 -f w 2 ) -j- iuu + vv 4- ivw'), if 2c« = 1 + - - - k and b = — K , 
and if we use this for Q„ we find that all necessary conditions are satisfied, including’ 
the condition for permanence notwithstanding collisions. 
19. If, however, the coefficients a , b were not the same for both colliding spheres, but 
the form were apcp + b 1 C)0C Z) + ~h by 3^]^3 fi - 3 *t~ &-C., then we should find that 
the coefficient of x'y in the new index is (1 — X 2 ) ciy + X 2 a 2 ; the coefficient of x 2 is 
(1 — X 2 ) a. 2 + X 2 c/, 1 ; the coefficient of x\x 2 is as before, b n ; but that of x\x 3 is 
(1 — X 2 ) h 13 + X 2 & 23 ; and that of x\x ?j is (1 — X 2 ) b. 23 -f- X 2 6 13 . 
The assumed law of distribution cannot in this case be unaffected by collisions, 
unless (l) all the a coefficients are the same; ( 2 ) if the velocities of the two colliding- 
spheres be Uy, Vy, iVy and n 2 ,v 2 ,w o, and those of any third sphere ber^ 3 , v 3 , iv 3 , then b 13 =b 23 , 
that is, the b coefficients must be such functions of the position, that if spheres 1 and 
2 are close together & 13 = b 23 , &c. 
To return to the case of the positions of the spheres being unknown. 
20 . If we form the determinant of the system 
we find 
D = (1 -f k) h — n -(1 + kY~ 1 
d u = (i + «)*-■ j(i + *)’-* 
Hence we find 
D„ 1 1 n -f k 
