6 G. V. L. Charlier 



Similar formulae are obtained for the increase in the number of stars between 

 the Hraits y ± \ Ay or z ± \ . Hence the whole increase in the number of stars 

 within the dominion Ail is found to be 



\ 'èx ■ dy d^, 



which thus represents the increase within AE caused by continuous changes in x, 

 y and 0. 



In hke manner we get for the increase caused by continuous changes of ?<, v 

 and w the expression 



.-. df , df , .df 



At AE\ii - - + — + i'v — 

 ^ "èu "dv dw 



where 



(3) 



du 



dh: 



1 



df 



~~ df'' 



m 



dv 



dhj 



1 





de 



ni 



dw 



d'^2 



1 



df 



~df 



m 



dt dt''' m ^ ' 

 denoting the components of the acting forces by X, Y, Z . 



2. These changes in the number of stars within the dominion AE are not 

 the only ones occurring in the time At. If in this time two stars pass each other 

 at a short distance, then the consequence will be that in the time At the velocities 

 of these stars — but not their coordinates — will undergo rather great, i. e. finite, 

 changes, whereas till now we have only considered continuous and small changes 

 in 'M, V, w. 



These discontinuous changes are of two essentially different kinds, which are 

 not kept apart in the kinetic theory of the gases. On the one hand there are 

 passages at a distance smaller than the mean distance B of the stars, on the other 

 we have real collisions. 



The number of passages, as well as the number of collisions, is proportional 

 to At AE and so also is the corresponding increase in the number of stars within the 

 dominion AE. We may therefore put the increase in the number of stars within 

 AE caused by passages 



= z(/ Ji? V (/) 

 and those caused by collisions * 



= AtAEU{.f)- 



Taking all these additions together and identifying them with 



AtAE% 

 dt 



