428 M. C. Szily on Hamilton's Dynamic 



them (for example, ~x = -^ocdt)\ then the equation to which 



Clausius in his answer appeals, and which he marked (18) in his 

 first memoir on this subject*, is the following: — 



£.fc+-^..^+S^-in+«r«ioBi. . . a) 



Now this is the equation to which Clausius refers as his own, 

 and which, he quite correctly remarks, has a more general appli- 

 cability than Hamilton's. It is also indeed impossible to find 

 complete accordance between this and Hamilton's equation. I 

 must, however, observe that I have not reckoned this equation as 

 Clausius's and designated it as such ; neither can I now so de- 

 signate it; for this equation was known five years since, as it 

 was given and demonstrated in the first volume of Sir William 

 Thomson's work, so widely diffused and frequently cited amongst 

 physicists, the ' Treatise on Natural Philosophy,' Oxford, 1 8t>7 

 (translated into German by G. Wertheim : Brunswick, 1871). 

 If, namely, the equation (9) in the section " Dynamical Laws 

 and Principles"! be written with Clausius' s notation, it exhibits 

 the following form : — 



Now it is clear that this equation is identical with equation (1) 

 above cited. 



Moreover, for a second and not less important reason I could 

 not, wheji speaking of the equation of Clausius, mean the above- 

 mentioned equation of Thomson : namely, Thomson's equation 

 is of such generality that, without a special hypothesis, it cannot 

 be brought into consonance with the second proposition of the 

 mechanical theory of heat. For if we could apply this equation 

 in its entire generality to that motion which we call heat, then 

 (as we shall see further on) the second proposition of the mecha- 

 nical theory of heat would not be true. We can, however, most 

 simply bring Thomson's equation into harmony with the second 

 proposition of thermodynamics, if we make a term vanish from 

 the dynamic equation, putting it =0. And because it was 



* Pogg. Ann. vol. cxlii. p. 442; Phil. Mag. S. 4. vol. xlii.p. 167. This 

 is equation (2) of his memoir " Ueber die Anwendung einer von mir auf- 

 gestellten mechanischen Gleichung " &c, Nachr. der kgl. Ges. d. Wiss. zu 

 Gbttingen, 1871, p. 248; Math. Ann. von Clebsch u. Neumann, vol. iv. 

 p. 232; Phil. Mag. S. 4. vol. xlii. p. 321. 



t In order fully to set forth the importance of this equation, Thomson 

 adds : — " This, it may be observed, is a perfectly general kinematical ex- 

 pression, unrestricted by any terminal or kinetic conditions." 



