.316 J. ~W. Gihhs on Graphical Methods in the 



pression occupying the place of an index to the sign 2* These 

 equations evidently include equation (8) as a particular case. 



It is eas}^ to form a material conception of these relations. If we 

 imagine, for example, mass inherent in the plane of the diagram with 



a varying (superficial) density represented by -, then 2 - 6 A will 



evidently denote the mass of the part of the plane included within 

 the limits of integration, this mass being taken positively or nega- 

 tively according to the direction of the circuit. 



Thus far we have made no supposition in I'egard to the nature of 

 the law, by which we associate the points of a plane with the states 

 of the body, except a certain condition of continuity. Whatever law 

 we may adopt, we obtain a method of representation of the thermo- 

 dynamic properties of the body, in which the relations existing 

 between the functions of the state of the body are indicated by a 

 net-work of lines, while the work done and the heat received by the 

 body when it changes its state are represented by integrals extend- 

 ing over the elements of a line, and also by an integral extending 

 over the elements of certain areas in the diagram, or, if we choose to 

 introduce such a consideration, by the mass belonging to these areas. 



The diflerent diagrams which we obtain by different laws of asso- 

 ciation are all such as may be obtained from one another by a process 

 of deformation^ and this consideration is sufficient to demonstrate 



* A word should be said in regard to the sense in which the above propositions 

 sliould be understood. If beyond the limits, within which the relations of v, p, t, e 

 and ?/ are known and which we may call the limits of the known field, we continue the 

 isometrics, isopiestics, &c., in any way we please, only subject to the condition that the 

 relations of v, p, t, e and ); shall be consistent with the equation dE=-tdT/—pdv, then in 

 calculating the values of quantities W and H determined by the equations d W=pdv 

 and dH=tdr] for paths or circuits in any part of the diagram thus extended, we may 

 use any of the propositions or processes given above, as these three equations have 

 formed the only basis of the reasoning. "We will thus obtain values of W and H, which 

 mil be identical with those which would be obtained by the immediate application of 

 the equations d W=pdv and dFI-=^tdr] to the path in question, and which in the case of 

 any path which is entirely contained in the known field will be the true values of the 

 work and heat for the change of state of the body which the path represents. "We 

 may thus use lines outside of the known field without attributing to them any physical 

 signification whatever, without considering the points in the lines as representing any 

 states of the body. If however, to fix our ideas, we choose to conceive of this part of 

 the diagram as having the same physical interpretation as the known field, and to 

 enunciate our propositions in language based upon such a conception, the unreality or 

 even the impossibility of the states represented by the lines outside of the known field 

 cannot lead to any incorrect results in regard to paths in the known field. 



