428 Lord Rayleigh on Dynamical Problems in 



Also 



$wf{w)dw=\{& -u) + u}f(w)dw 



=K»-«)V(u)+t(»^) , jf , (u)+. . . 



+ uf(u){wf(u) + i(w-u¥f(u)+ . . ,}; 

 so that 



\ wf(w) die = uf(u) . (u" — u') 



+ {i/W+i"/Wl|(»"-») 2 -(»'-»)V' • • 



= r=T"/(«) + |^ 2 </(») +«/>)} + cube S of j. (12) 

 As far as (/ 2 inclusive (10) thus becomes 



or 



If^ 1 = ^ J ^ disappears from the first term as it stands, and 

 will do so in any case in the limit when it is made infinitely 

 small. Moreover, in the second term u 2 is to be neglected in 

 comparison with v 2 . We thus obtain 



«/(«){l + «M}+^ 2 /'(«)=0 • • • (13) 



as the differential equation applicable to the determination of 

 f(u) when q is infinitely small. The integral is 



qv 2 log/ (u) + J(l + Vi/v)u 2 = constant, 



or 



f(u)=Ae- hu \ (14) 



where 



»-^> <m 



or. if i?i=«, 



h=l/qv 2 (16) 



The ultimate distribution of velocities among the masses is 

 thus a function of the energy of the projectiles and not other- 

 wise of their common mass and velocity. The ultimate state 

 is of course also independent of the number of the projectiles. 



