310 rl. W. Glhhs on Graphical Methods in the 



dW^=. apdv, (:i) 



de= fidH—d W, (b) 



d,/= __ ^ (c) 



where a and /i are constants depending upon the units by which y, f>, 

 JV and Hare measured. We may su})pose our units so chosen that 

 a=il and /iz^ljj- and write our equations in the simpler form, 



dE = dII—dW, (1) 



dW = pdv, (2) 



dll=^ tdf]. (a) 



Eliminating d W and <///, we have 



^f = tdif — ^xfe. (4) 



The quantities y, ji>, ?, f and ?^ are determined when the state of the 

 body is given, and it may be permitted to call thanx functions of the 

 state of the body. The state of a body, in the sense in which the 

 term is used in the thermodynamics of fluids, is capable of two inde- 

 pendent variations, so that between the five quantities t% />, ^, f and ;; 

 there exist relations expressible by three finite equations, diflerent in 

 general for diiFerent substances, but always such as to be in harmony 

 with the differential equation (4). This equation evidently signifies 

 that if f be expressed as function of v and /;, the partial differential 

 co-efficients of this function taken with respect to v and to // will be 

 equal to —p and to t resj^ectively.J 



* Equation (a) may be derived from simple mechanical considerations. Equations 

 (b) and (c) may be considered as defining the energy and entropy of any state of the 

 bod}", or more strictly as defining the differentials dt and dn]. That functions of the 

 state of the body exist, the differentials of which satisfy these equations, may easily 

 lie deduced from the first and second laws of thermodynamics. The term entropy, it 

 will be observed, is here used in accordance with the original suggestion of Clausius, 

 and not in the sense in which it has been employed by Professor Tait and others after 

 his suggestion. The same quantity has been called by Professor Rankine the Thermo- 

 dynamic function. See Clausius, Mechanische Warmetheorie, Abhnd. ix, §14; or 

 Pogg. Ann., Bd. cxxv (1865), p. 390; and Rankine, Phil. Trans., vol. 144, p. 126. 



f For example, we may choose as the unit of volume, the cube of the unit of length, 

 — as the unit of pressure the unit of force acting upon the square of the unit of 

 length, — as the unit of work the unit of force acting through the unit of length, — and 

 as the unit of heat the thermal equivalent of the unit of work. The units of length 

 and of force would still be arbitrary as well as the unit of temperature. 



\ An equation giving e in terms of i] and v, or more generally any finite equation 

 between e, rj and v for a definite quantity of any fluid, may be considered as the funda- 

 mental thermodynamic equation of that fluid, as from it by aid of equations (2), (3) and 

 (4) may be derived all the thermodynamic properties of the fluid (so far as reversible 



