ON ODR KNOWLEDGE OF THERMODYNAMICS. 89' 



about their mean values, and there shall be one or more ' quasi-periods,' 

 i, satisfying the definition which will be given in the course of the proof 

 (equation 13, infra). 



(ii.) When the state of the system is changed (as by the communica- 

 tion of heat or by changes in the volume or external configuration of a 

 body), such changes shall be capable of being treated as small variations 

 of the motion from the state of steady motion. 



Helmholtz, in his paper on Monocyclic Systems, makes a similar as- 

 sumption — namely, that the changes in the state of the system shall take 

 place so very slowly that the motion of the system at any instant differs 

 infinitesimally little from a possible state of steady motion. Tliis is the 

 exact equivalent of the assumption always made in treating the Second 

 Law from a physical point of view — namely, that heat is communicated 

 to or taken from the working substance so slowly that at every instant 

 of the process the temperature of the body is sensibly uniform through- 

 out. 



12. With these assumptions, let the positions of the molecules be 

 determined in the first instance by the Cartesian coordinates (*, y, z) of 

 the particles (iii) forming them. 



Suppose that the state of the system also depends on the values of 

 certain other coordinates, p^, p.2, &c., which, as suggested by J. J. Thom- 

 son,' we shall call the 'controllable coordinates' of the system; to this 

 class belong the volume of the body, the charge of electricity present 

 on it, or any coordinates which can be acted on directly from without. 

 The values of these latter coordinates will enter into the expression for 

 the potential energy of the system. 



Let T=kinetic energy of sjstem=^^^in{x'^ + y- 4- P) . 



V=potential energy. 

 E=total energy=T-FV. 



In Thomson and Tait's ' Natural Philosophy,' part i. § 327, it is 

 shown that 



8 {''2Tdt=^^m{iSx + y8y + ^Sz)T+ NsT-^mixBx + ySy + zsAdt (5) 



But by D'Alembert's Principle we always have for the motion of the 

 system 



whence 



^m(x8x + y8y + zSz)=-'^(^^^8x + ^l8y+'^^^8z^ . . (6) 



Now, V is a function not only of the molecular coordinates {x, y, z) 

 but also of the controllable coordinates 2h> P-2i ■ • • ^^^ these latter are 

 also liable to variation. Hence for the complete variation of V we have 



' Applications of Dynamics to Physics and Chemistry, p. 94. 



