Illustration of the Theory of Gases. 435 



sharp contrast between the steady state and the early stages 

 of the variable state. The latter depends upon the momentum 

 of the projectiles, and upon the number of impacts ; the 

 former involves the energy of the projectiles, and is independent 

 of the rapidity of the impacts. 



The mean square of velocity after any number (n) of 

 impacts is 



J+co 

 u 2 f(uj ?i)du = n, 

 -co 



or, if we restore 4<^ 2 y 2 , 



mean u 2 = n Aq 2 v q (27) 



It must be distinctly understood that the solution expressed 

 by (24), (25), (26) applies only to the first stages of the 

 bombardment, beginning with the masses at rest. If the 

 same state of things continued, the motion of the masses 

 would increase without limit. But as time goes on, two 

 causes intervene to prevent the accumulation of motion. 

 When the velocity of the masses becomes sensible, the chance 

 of an unfavourable collision increases at the expense of the 

 favourable collisions, and this consideration alone would pre- 

 vent the unlimited accumulation of motion, and lead to the 

 ultimate establishment of a steady state. But another cause 

 is also at work in the same direction, and, as may be seen 

 from the argument which leads to (13), w r ith equal efficiency. 

 The favourable collisions, even when they occur, produce less 

 effect than the unfavourable ones, as is shown by (6) and (7). 



We will now investigate the general equation, applicable 

 not merely to the initial and final, but to all stages of the 

 acquirement of motion. As in (20), (23) we have 



df(u,t)du _ vdt rl j., , N , ,. , , 

 - dt dt = — {i/O) • (v-u')du' 



+ i/O") • (v+u")du"-f(u) . vdv] ; 

 and thus by the same process as for (22) 



&-*£ww>+V*a5- . • .(28) 



If we write, as before 



we have 



t' = 2q*vht, zndh = l/qv 2 , .... (29) 



%= d j4+2hUuf) (30) 



dt du 2 du K J J 



2 G2 



