ON OUR KNOWLEDGE OF THERMODYNAMICS. 103 



system formed by coupling (1) and (2), the geometrical equation (34) 

 gives 



iL= 2a (53) 



where *'(si) is a function of s, alone and ^'(sj) is a function of Sg alone 

 Therefore, comparing (53) and (54), we must have 



dF dF 



Putting 

 (55) gives 



. (55) 



Hs,)=.\^'(s,)ds„ n^o)=\^'(s,)ds, . . (56) 





(57) 



The integral of this can be written in the form 



X(F(s,S2))=X(o-)=$(si)+*(«2) + . . (58) 



where X denotes any arbitrary function of F or cr. 



Equation (58) determines the general form of the quantity correspond- 

 ing to entropy in the system formed by coupling the two monocyclic 

 systems (1) and (2) in a manner satisfying the conditions of the present 

 problem. 



Moreover, in the individual systems we have by (56) 



t ... (59) 



so that the quantities q\/^'(^i) ^^^ 22/^(^2)) which are equated when 

 the systems are coupled, are integrating divisors of dQ^ and tZQ,. This 

 kind of coupling is therefore 'isomorous,' and is analogous to the thermal 

 contact of bodies at the same temperature. 



29. Thus Helmholtz has shown that all the thermodynamical pro- 

 perties of matter can be represented dynamically by means of monocyclic 

 systems which are capable of being coupled together. In coupling such 

 systems it has been assumed that — 



(i.) The forces acting on the controllable coordinates are unaffected, 

 so that only the motions of the molecular or gyrostatic coordinates are 

 connected together, and the coupled system is monocyclic. 



(ii.) The geometrical equations connecting the two systems can be put 

 in the form ^i=i/'2, so that </>, and 1/^2 possess the same properties which 



