102 On some Questions in the Kinetic Theory of Gases. 



The number of molecules of the N, molecules which, during 

 the time Bt, come into collision with any others, is obtained 

 by integrating the expression given by equation (11) for dn 

 with reference to all possible values of du, dv } die, and d\. 

 The result of this integration may be written thus : — 



N 3 = do dco crH^ffV cos 6 dp d\. . . . (25) 



Besides these N 3 molecules, all the molecules whose number is 

 denoted above by N 1} after lapse of the time St, reach the 

 parallelepiped do*, and their velocity-points the parallelepiped 

 do*. 



Of the molecules whose centres are in the parallelepiped do 

 there are, however, during the time St some (let their number be 

 N) which had before altogether other velocities, but by impact 

 with other molecules have attained exactly such velocities that 

 their velocity-point after impact lies in the parallelepiped dco*. 

 We have therefore N 2 = N! — N 3 + N 4 . Observing equations 

 (23), (24), and (25), and remembering that N 4 was obtained 

 from equation (16) by integration exactly as N 3 from equa- 

 tion (11), we obtain, since, as already remarked, do*— do and 

 day* = dco, 



d* d# dy on B? (h d£ 



=cr 2 ^ (f A' - ff i)y cos edpdx. . (20) 



The right-hand side of equation (22) is a sum of terms of 

 which no one can be positive. But since, as soon as the gas 

 is at rest and does not change its condition, the left-hand side 

 must vanish, so also each term of the right-hand side must 

 vanish by itself ; and we have generally for any values what- 

 ever of the variables ff x =ff\. 



Since this equation must hold for all possible values of the 

 variables occurring in it, so long as only the condition 



P+V + € 3 =f a +V a +? ra .... (27) 



is fulfilled, it follows at once in the usual way that /', if the 

 gas be at rest, must have the form 



F^-^+f+S 8 ), (28) 



where h is a constant, and only F involves the variables x, y, z. 

 In order to determine F we make use of equation (26). We 

 see at once that since / has the form (28), the expression 

 f'fi'—ffi must always vanish, since equation (27) must be 

 satisfied for every impact. Therefore also the right-hand 

 side of equation (26) must vanish, and the substitution of the 



