342 Graphical Methods hi Thermodynamics. 



The arrangement of the isometric, isopiestic, etc., as given in figure 

 13, will indicate directly the sign of any differential co-efficient of the 



form I — — I , where w, w and z may be any of the quantities v, 2>, t, if 



(and 5, if the isodynamic be added in the figure). The value of such 

 a differential co-efiicient will be indicated, when the rates of increase 

 of V, p, etc., are indicated, as by isometi'ics, etc., drawn both for 

 the values of v, etc., at the point A, and for values differing from these 



by a small quantity. For example, the value of (-y-) will be indi- 

 cated by the ratio of the segments intercepted u))on an isentropic by 

 a pair of isometrics and a pair of isopiestics, of which the differences 

 of vohmie and pressure have the same numerical value. The case in 

 which W or i^T appears in the numerator or denominator instead of a 

 function of the state of the body, can be reduced to the ]jreceding by 

 the substitution of pdv for d TF, or that of tdi] for dll. 



In the foregoing discussion, the equations which express the funda- 

 mental principles of thermodynamics in an analytical form have been 

 assumed, and the aim has only been to show how the same relations 

 may be expressed geometrically. It would, however, be easy, starting 

 from the first and second laws of thermodynamics as i;sually enunciated, 

 to arrive at the same results without the aid of analytical formula^, — to 

 arrive, for example, at the conception of energy, of entropy, of abso- 

 lute temperature, in the construction of the diagram without the ana- 

 lytical definitions of these quantities, and to ol)tain the various prop- 

 erties of the diagram without the analytical expression of the thermo- 

 dynamic properties which they involve. Such a course would luxve 

 been better fitted to show the independence and sufficiency of a graphi- 

 cal method, but perhaps less suitable for an examination of the com- 

 parative advantages or disadvantages of different graphical methods. 



The possibility of treating the thermodynamics of fluids by such 

 graphical methods as have been descriljed evidently arises from the 

 fact that the state of the body considered, like the position of a })oint 

 in a plane, is capable of two and only two independent variations. 

 It is, perhaps, worthy of notice, that when the diagram is only used 

 to demonstrate or illustrate general theorems, it is not necessary, 

 although it may be convenient, to assume any particular method of 

 forming the diagram; it is enough to suppose the different states of 

 the body to be represented continuously by |)oints upon a sheet. 



