Illustration of the Theory of Gases. 427 



denoted hjf(u)du } the problem before us is the determination 

 of the form oif(u). 



The number of masses, whose velocities lie between u and 

 u + du, which undergo collision in a given small interval of 

 time, is proportional in the first place to the number of the 

 masses in question, that is to f (u) du, and in the second place 

 to the relative velocity of the masses and of the projectiles. In 

 all the cases which we shall have to consider v is greater 

 than u, so that the chance of a favourable collision is always 

 proportional to v — u, and that of an unfavourable collision 

 to v -f u. It is assumed that the chances of collision depend 

 upon u in no other than the above specified ways. The 

 number of masses whose velocities in a given small interval 

 of time are passing, as the result of favourable collisions, 

 from below u to above u, is thus proportional to 



I 



f(w) . (vi— w) dw, . . . . . (8)' 



where u is defined by (6) ; and in like manner the number 

 which pass in the same time from above u to below u, in con- 

 sequence of unfavourable collisions, is 



%J u 



f(iv).(y 1 + w)div, (9) 



u" being defined by (7). In the steady state as many 

 must pass one way as the other, and hence the expressions 

 (8) and (9) are to be equated. The result may be written, in 

 the form 



«x { - \f(io)dw= , 



^* J It' J U •* J u' 



wf\w)dw. . . (10) 



Now, if q be small enough, one collision makes very little 

 impression upon u ; and the range of integration in (10) is 

 narrow. We may therefore expand the function /by Taylor's 

 theorem : — 



f(w)=f{u)+(wru)f'(u)+±(w-uyf\u)+ . . .; 

 so that 



| P - P"} / (w)dw = (2u-u'-u")f(u) 



= - I^/M - (1^)2 (^ 2 + u ") /' M + cubes of 9- (11) 



* In the present problem i\=v, but it will be convenient at this stage 

 to maintain the distinction. 



