40 Prof. R. Clausius on the Theorem of the Mean Ergal, 

 which can also be written in the form 



For the expression on the left-hand side we introduce the symbol 

 £ putting 



. . . . .'-i-Wi (45) 



so that the preceding equation is transformed into 



b= 1 - \fl - 2 — - P(hc«) ' « -f n — (3 2 {kc n ) ~ I . (46) 



This equation, in which f appears as a function of kc n 3 we again 

 make use of in order, conversely, to exhibit kc n as a function of 

 f, remarking that f, as is readily seen from the preceding equa- 

 tion, is a small quantity of the order of /3, and therefore, in the 

 development of a series according to ascending powers of f, the 

 first terms only need to be taken into consideration. We get 



After attaining this, we divide the last of equations (30) by 

 the first, thereby producing 



_ 2£ ,!LL?/ rc-2/3 * , \ 



. . hex -r—k » H -£ »+...), 



» c ■ ■• \ w c / 



or, written differently, 



2 n-ir n o * "1 



hc n = £ £(**») T 1 1 + —^ £(&») " » + . . . J . 

 Putting in this for kc n the expression given in (47), we get 



We can therefore express the sought-for result in the following 

 manner : — If % is a function the form of which is determined by 

 the equation 



*»-®-o?r[> + s=i,c + -...]..-<*. 



then is 



Simultaneously, by the employment of (47), we obtain from 



