44 Prof. li. Clausius on the Theorem oj the Mean Erya! } 



Having made these preliminary remarks, we will form the 

 differential coefficients which are to be compared. For this pur- 

 pose we first consider h as a function of u and c } and form the 

 differential equation 



7 d c h , d u h 7 

 ah — -j~ du-\ — 7- dc. 

 du dc 



Herein let us now imagine u represented as a function of w and 

 c ; we can then write the equation thus : — 



,d u h 

 + -j-dc 

 dc 



77 dji fd c u , . d w u 7 \ 

 dh= -^l-^-dw+ -^-dc) 

 du \dw dc / 



/dji d^ d u h \ , 

 \ du dc dc / 



d c h d c u 

 du dw 



If, on the other hand, we regard h as a function of w and c } we 

 can write 



dh^^dw+^dc. 

 dw dc 



Now, as in this and the preceding equation the coefficients of 

 dw, and in like manner the coefficients of dc, must be equal to 

 one another, we cret 









d c h d c u 

 du dw 



d c h 

 dw 











dji d w u 

 du dc 



dji 

 dc 



_d w h . 

 dc 



or 



by 



trail 



sposition 



dji 

 d c h dw 

 du d c ii 



dw 





i 









dji d w h 



dc dc 





dji d w u 

 du dc ' J 



(55) 



We will give these equations a form somewhat different and 

 more adapted for our use, putting 



coefficient; for in his " Berichte " (p. 441) he denotes a differentiation ac- 

 cording to x in which y is considered constant by — — . Although this 



dxy 

 notation is convenient, yet I think the one I have chosen may be retained, 

 because the index is not intended to specialize the variable in the denomi- 

 nator, but the nature of the differentiation, and therefore, in my opinion, 

 is better placed close to the d (which denotes the differentation) of the nu- 

 merator. 



