4 



C. V. L. Chiirlier 



The first problem in statistical mechanics is to deduce a differential equation 

 for the function /. 



The formula (1) gives the number of stars, which at the time t are situated 

 within the dominion Ail Ab = AE. The number of stars, which at the time t -f At 

 are situated within the same dominion AE is then 



A -j- = f{f + ^> 



= X, y, 0; + 



Even At may not be chosen as small as we please. How small can not be 

 determined until later. 



The increase of stars within AE in the time At is thus: 



(2) AA = At^ AiiAs. 



^ ' dt 



There is, however, another method for calculating AÄ, namely through directly esti- 

 mating the number of stars wandering out from APJ or in this dominion. 



The examination is made easier by a consideration of each one of the variables 

 X, y, u, V, w separately. The number of stars which simultaneously pass the limits 

 for two of the variables is, indeed, of the second order regarding At, which order 

 as well as all higher ones may be neglected. 



Let us first examine the outcome of contimious variations in x, y, u, v, w 

 upon the increase in the number of stars within AE . 



The six variables may be represented geometrically by six (straight) lines 

 Consider the x — hne. 



X 00 VC 



Fig. 1. 



Denote the limits of x, within AE, by x' und x", so that 



and 



x' -\ x" 



X = 



2 



The number of stars between these limits is according to (1) 



A = f{x) {x" — x) Ay Az As. 



■j [x" — x) Ay As Ab. 



All stars having a value of x between x and x" are hence supposed to 

 possess the same values of y, z, u, v, w, or, more strictly, values of these variables 

 within AE. 



