38 Prof. It. Clausius on the Theorem of tlie Mean Ergal, 



If in the last two of the equations (30) we insert the expression 

 for k 2 hence arising, and also the expression which according to 

 (31) holds for i, 



n+2 



we obtain from them the following : — 



a,= c -»f-=[l-2^i/3JS + (^=?) V{5+ . . .]; (33) 



A= c -f-»g / 8p-H(5=?)>p +...]. . . (34) 



And by subtracting the first of these two equations from the 

 second we get, further, 



^-w=c-f - 2 [-l +2/3f«-^^)V~F+ • • •] • (35) 



If, lastly, in this we substitute for £ again its value given in 

 (31), we obtain k — w represented as a function of i. Here, 

 however, we will not actually carry out this substitution, but, in 

 order to obtain short formulae, express the result thus: — If cf> 

 denotes a function the form of which is determined by the equation 



<K?)=r 2 [-l+2/3|f»-(^)>^+ !..], (36) 

 then is 



h _ w = c -n^_i_ c -^ (37) 



At the same time, it follows from (33) that we can put 



. . (38) 



,„-L- w& #Jg)-L-,, # fe c "" 2 ) 



2 C * d% ~2 C l —Ji —' 



and therefore also 



1 .d{h-w) 



It is still easier to represent k as a function of e. For since 

 the signification of e is determined by the equation 



fdx\ 

 e — m [-Ji) i = 2mwi, 



we need only multiply the second of the equations (30) by 2 in 



