﻿Deduction of Thermody namical Relation*. 755 



being chosen in the positive direction of the coordinates. 

 The entropy equation is then 



du=Hds + Adct + Bd0 + .... 



As before, we can transform this equation by subtracting 

 from both sides one or more expressions like d(Ts), d(A.x), &c. 

 As the second side has n + 1 terms, it is easily seen that 

 altogether 2 M+1 equations may be written down) the above 

 one included, each new form giving rise to the introduction 

 of a thermodvnamical function, so that including u there are 

 2 n + l of these. 



The thermodynamic^ relations which follow from these 

 equations are not all of the same kind : some of them involve 

 the first term (Tds or sdT), others are between the remaining 

 terms {Kdot or adA, &c.) mutually. The total number of 



Inch 



relations deducible from each equation is — — ——, of 



(n — l)n ■*■ ' * 



— z. — ~ — are between the forces and coordinates with either 



of the two conditions s = const. or T = const. attached (these 

 we shall call the elastic relations), and the remaining n 

 (analogous to Maxwell's relations) containing s and T as 

 variables. The total number of relations of the form con- 

 sidered is therefore 2 n n(n + 1) involving 2 n - 1 (n — l)>iadiabatic 

 elastic relations, an equal number of isothermal elastic rela- 

 tions, and 2 n+1 n of the Maxwell pattern. 



It deserves notice that those relations in which amongst 

 the constant quantities no coordinates occur such as the 

 elastic relation 



3* _ d/3 



BBtacd... c)Atbcd... ? 

 or the Maxwell relations 



d' = __df__ and ^ T = - _1°L_ 



BATBCD... 9TABCD... dA. 5 RCD... "dskBCD... 



are usually of more importance than the others, because as a 

 rule experimentally forces are more easily kept constant than 

 coordinates. It may also be remarked that in many problems 

 which deal with only one or a few of the coordinates while the 

 others are simply ignored, the relations which are obtained 

 have all to be taken as involving the condition that these 

 others, but more often the corresponding forces, remain 

 constant. 



3 D 2 



