Principle in Thermodynamics. 429 



Clausius who showed which terra must be omitted from that 

 equation, and under what assumption this can be attained; I 

 think that the complete truth of the historical development is 

 attained when we, as I have done, designate as Clausius's the 

 special equation* resulting from the assumption above alluded 

 to, viz. 



SU = ST+2T81ogi (3) 



Now, what is the difference between this and Hamilton's 

 equation ? 



Before entering further into the clearing-up of this question, 

 let us inquire generally what kind of conditions must be fulfilled 

 in order to bring Thomson's equation into agreement with the 

 second proposition of the mechanical theory of heat. For the 

 sake of a more convenient survey, I will write Thomson's equa- 

 tion in a somewhat different form. If E denote the total energy, 

 therefore the sum of T and U, we can, after a slight transforma- 

 tion, write equation (2) as follows : — 



or, if we restore the definite integral in the last term, from which 

 the mean value denoted by the horizontal stroke has arisen, we 

 obtain 



If, finally, we divide the entire equation by T, it becomes 



5E „ _1 C * f~*TT /dV - dV 9 dUV 1 JA 



■, T -=S21og( ! -T) + ^J o [SU-(^S, + - (¥ 8, + ^)8,]*. 



It is plain that this equation, which is still identical with 

 Thomson's, can only be brought into the form of the second 

 proposition when it is possible to put the last term equal to the 

 total variation of any invariable function of the coordinates — 

 that is, when 



i>-(f & +f^§ *)]*-^ • m 



Supposing this value introduced into the last equation, and 

 the symbols of the differentials substituted for those of the 

 variations, then 



-==d2hg{W)+df. 



* This is Clausius's equation (21) in Pogg. Ann. vol. cxlii. p. 442; 

 Phil. Mag. S. 4. vol. xlii. p. 170. 



