432 Lord Rayleigh on Dynamical Problems in 



The masses which by single collisions at velocity v would 

 ultimately produce the same effect as these pairs are therefore 

 very approximately 2q. 



If the projectiles be distributed in pairs in such a way that 

 the components of each strike nearly simultaneously and upon 

 opposite sides, 



1 — qi^l — q 1 — q J 1 — q 



{l-qY U± {1-qf 

 _l + 2^ + (^ 2 ) Aq.qv 



~ l~2q + (q 2 ) U± l-2q + (q 2 ) ; 



showing that the effect is the same as if the mass were 

 doubled, and the velocity reduced from v to qv. Thus, when 

 q is infinitely small, the effect is negligible in comparison 

 with that obtained when the connexion of the components of 

 a pair is dissolved, and each individual is projected at random. 



Another Method of Investigation. 



The method followed, in the formation of equation (10) 

 seems to lead most simply to the required determination of 

 f (u) ; but it is an instructive variation to consider directly 

 the balance between the numbers of masses which change 

 their velocities from and to u. 



The number of masses whose velocities lie between u and 

 u + du being f(u)du, we have as the number whose velocities 

 in a given small interval of time are expelled from the range 

 duj 



f(u) du (v — u) +f{u) du (v + u) , 

 or 



2vf(u)du. 



This, in the steady state, is equal to the number which enter 

 the range du from the two sides in consequence of favourable 

 and unfavourable collisions ; so that 



/ (V ) (« - u') du' +f (u") (v + u") du" - 2vf(u)du = . (20) 

 By (6), (7), since v is constant, 



du'= z. -du, du" = = du : 



1 — q 7 i~q 



so that 



S/w • (•-«■) + i3a b ">- (»+«")-2«/(«)=o. 



