and its Application to the Molecular Motions of Gases. 37 



neously that the scries in (27), which is the difference of the 

 two others, can be represented by 



n n 



Inserting these expressions for the respective series, we will 

 introduce a single symbol for the product aR which occurs in 

 all three equations, by putting 



£ = aR (29) 



The equations thence arising, after being multiplied by 4, are : — 



<-=V!(<-^V;); 



Wl 



i=2</2k{c-0k »); 



4 - '' 

 hi= - ,/2/3/c 



(30) 



§ 6. We will now employ these equations in order to repre- 

 sent the quantity h—w as a function of i } the quantity k as a 

 function of e } and h as a function of u. 



For this purpose, in the first place we multiply the first of the 

 1 »+g 

 equations (30) by c 2 ; and then, for the sake of abbre- 



viation, denote by a single letter the product which will stand 

 on the left side, putting 



?== — 7-c~^T (31) 



2^2 V ; 



The equation can then be written in the following form : — 



f=(# c »)-iri_. !Ljj^£(fo^iJ. .... (32) 



In this equation the quantity f is represented as a function of 

 Jcc 11 ; and it may be remarked that the second term within the 

 square brackets is very small in comparison with 1, on account 

 of the factor /3. This circumstance makes it easy, conversely, 

 to represent kc n or even its square root as a function of f, by 

 forming a series arranged according to ascending powers of /?, 

 in which series only a few terms need be taken into account. 

 Thereby is produced the following equation : — 



»» t S 1 n-% » ^(n-2) 2 *, 1 



