2 GRAPHICAL METHODS IN THE 



These are subject to the relations expressed by the following differ- 

 ential equations : dW=p&>, (a) 



de = pdH-dW, (b) 



, dH* 



d n= , (c) 



where a and /3 are constants depending upon the units by which v, p, 

 W and H are measured. We may suppose our units so chosen that 

 a = l and /3=l,t and write our equations in the simpler form, 



de = dH-dW, (1) 



dW=pdv, (2) 



dH=tdtj. (3) 



Eliminating dW and dH, we have 



de i= F<## "p dv. (4) 



The quantities v, p, t, e and i\ are determined when the state of the 

 body is given, and it may be permitted to call them 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 v, p, t, 6 and r\ 

 there exist relations expressible by three finite equations, different in 

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

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

 that if e be expressed as function of v and rj, the partial differential 

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

 equal to p and to t respectively. { 



* 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 body, 

 or more strictly as defining the differentials de and d-rj. That functions of the state of 

 the body exist, the differentials of which satisfy these equations, may easily be 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 Wdrmetheorie, 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 c in terms of TJ and v, or more generally any finite equation 

 between e, i\ 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 

 processes are concerned), viz. : the fundamental equation with equation (4) gives the 

 three relations existing between v, p, t t e and rj, and these relations being known, 

 equations (2) and (3) give the work W and heat H for any change of state of the fluid. 



