430 M. C. Szily on Hamilton's Dynamic 



If we integrate this equation throughout an entire cyclical 

 process, taking at the same time into consideration that »T as 

 well as F (according to the assumption) has at the beginning 

 and at the end of the process the same value, then is 



I 



^=0 



rn — V7 » 



Thomson's equation is therefore only then in complete accord- 

 ance with the second proposition of the theory of heat, when the 

 equation of condition (equation 6) is satisfied. 



Let us now see in what manner this condition is fulfilled, on 

 the one hand^ in Hamilton's, and, on the other, in Clausius's 

 equation. 



Hamilton founded his equation on the assumption* that the 

 form of the force-function does not change with the change of 

 the motion, that hence SU depends solely on the variation of the 

 space-coordinates — that is, that 



8 »-(f^f *+£*)^ • • « 



In consequence of this, 



SF = 0, 



and equation (5) takes the following form : — 



U«i£*k ....... (8) 



This (8) is Hamilton's equation. 



Now, in what consists the assumption made by Clausius ? He 

 admits that even the form of the force-function may change ; he 

 only lays down this condition f, that the variation which results 

 from the change of form of the force-function, taken at its mean 

 value, is equal to 0, and therefore 



dx dy y dz ' 



that is, 



B 



»-("> + f s ' + '»>-«• ■ i?) 



dy 



Therefore according to this assumption also 



SF=0. 



Consequently equation (1) takes the form of equation (3), viz. 



8U = 8f +2T81ogi. 



* Phil. Trans. 1834, p. 249, sub (1). 



t Pogg. Ann. vol. cxlii. pp. 442-446, sub (18)-(22) ; Phil. Mag. S. 4. 

 vol. xlii.pp. 167-170. 



