Distribution e-' 2h x and van der Waals' Theorem. 411 



m. Let m undergo its first collision at time t after the given 

 instant, and let that change u into u + u f . We might then 

 calculate the chance of t and v! having in that haphazard 

 distribution given values. And so we could find the mean 



11 

 value of — , as a function of h and o£" the coordinates 



■ u' 



^i?/i£i&c., an d x y z. And if, — being so calculated, we 



make -77- = ~r -> 77 is a function of h and of the coordinates 

 at t dt 



of all the spheres, including x y z. The like is true o£ 



-7* and — -. And as it is true of m, so also for every other 



did 



sphere, the mean values of -7- - -=7 , -7- in this distribu- 



dt dt 7 dt 



tion are determinate functions of h and the coordinates. 



With this meaning of -77 , -77, -77 , let p be a function of 



x y z which at the instant considered satisfies for every 

 molecule the conditions 



* dx dt 



om rfp dv . 



^Ty = - m dt> 

 . om dp dw 



and ^S^pdxdydz throughout S =N. 



Then p is at the given instant a determinate function of 

 xyz. 



Now let us use the same law of distribution of the co- 

 ordinates and velocities as in Boltzmann's theorem, namely, 



In stationary motion we have, as in art. 2, summing for all 

 the spheres 



^(c'x d , du d\ . 7V 2 ■ 

 \dt dx dt du/ 



and substituting its value for -7-, 



«i£ + *"i , *py~ t ~°> 



