1891.] 



On the Collision of Elastic Bodies. 



Ill 



coordinates pi . . . . p r , and another set called systems m with co- 

 ordinates p r +\ . . . . p n - Let F(pi . . . .p r Vi . . . . v r )dpi . . . . dv r or 

 F . dpi . . . . dv r be the number of systems M with coordinates and 

 velocities between 



and pi + dp, 

 Ac., 



and V 

 &c., 



(A), 



dv u the corresponding num- 

 , &c .............. (B). 



and f(pr+i Pn v r+ i .... v )t )dp r +i . 

 ber for the other set, between limits 



p r +i and 



Let YT be a function, such that when -^ = a collision occurs. 

 Then d\^jdt or R denotes the frequency of collision. And 

 F/. E, . dpi . . . . dv n -\ denotes the number in unit of time of collisions 

 between members of the two sets having their coordinates p x . . . .p n -i 

 and velocities v\ . . . . v n . 



Similarly, the number in unit time of collisions in the reverse di- 

 rection is 



F'/ R dp\ ____ dp' n _! dv l ____ dr,,_a dR. 



In the Maxwell-Boltzmann distribution F/, F'/' are functions of the 

 kinetic energy only, and this being the same in the two states, 

 F/= F'/'. And as many direct as reverse collisions take place in unit 

 time, which insures the permanence of the distribution. 



12. If F/ ^t F'/'. then the number of systems of the first kind whose 

 coordinates and velocities lie between 



Pi and 



Ac., 

 ^i and Vi -\-dvi, 



&c., 



is increased per unit of time by collisions 'with the second set, having 

 coordinates and velocities between 



and 



&c., 

 v r +i and Vr+i+dvr+i, 



&c., 

 by the quantity 



dp ; . . . . dv r (F'/'-F/) R dp r+ i .... dvn^i dU, 



and by collision with systems m without restriction by the quantity 

 dp,. . . . dv r JJ . . . . (F'/'-F/) R dp r+l . . . . dv n ., dR, 



