38 



C. V. L. Charlier 



The first of these integrals does, however, vanish, because f\ = 0 for = ± oo , 

 as well as for = ± co or *f ^ = ± co . 



Substituting the values of (2), (3) and (6) we now get 



dt ^ dx ^ dy ^ du^ di\ dw^ 



+ (4) + (5). 



Here Q may denote any function of t; x, y, 0; t<j^, v^, w^. 



If, especially, Ç = log/^, we get the Ä- function of Boltzmann 



(62) H = j\ogf^f,ds^d9.. 



The formula (61) now gives 



dH 

 df 



dz^ dÜ 



+ (4) + (5) . 



The integral in the right member of this equation has, however, the value zero. 

 We have indeed 



X 1^ rfs, dil = 



de, d9. = 0. 



Put for a moment 



/ = If dB dQ, 



the integral taken over the whole dominion il , then we have 



dl 



dt 



^^ds d9.+ {/dB d9 



or, according to (60) 



dl C df , df , df 



o 



But ] is nothing but N= the total number of stars, which is not altered with 

 the time. Consequently we get 



dt 



so that now the expression for dH : dt takes the form 



(63) ^ ^ 1^ ^^^'^^ '^'^ + 1° ^-^'^ "^'^ 



29. We have, hitherto, taken the collision-function also into consideration, 

 though it does not occur in the Jï-theorem, which is not concerned with real (une- 

 lastic) collisions, but only with passages. In the following the collision-function is 

 left out, so that 



