N4 



332 J. W. Gihhs on Graphical Methods in the 



have l)C'en discussed in detail in tliis article. It is easy to find an 

 expression for y depending simply upon the variations of the energy, 



by comparing the area and the work or 

 7) heat of an infinitely small circuit in the 



form of a rectangle having its sides 

 parallel to the two axes. 



Let N 1N2N3N4 (fig. 8) be sxich a cir- 

 cuit, and let it be described in the order 



' I 1 '^ of the numerals, so that the area is posi- 



I ^ tive. Also let f ,, fo? ^31 ^i I'cpi'csent 



^ the energy at the four corners. The 



work done in the four sides in order 



commencing at Nj, will be s^ — eoi^i 



63 — 64, 0. The total work, therefore, 



~Y for the i-ectangular circuit is 



^-^3 + ^3-^4- 

 Now as the rectangle is infinitely small, if we call its sides dv and f?//, 

 the above expression will be equivalent to 



d^e -, , 



5 — z- dv dri. 



dv dtj 



Dividing by the area dv di], and writing y^ for tlie scale of work and 



heat in a diagram of this kind, we have 



1 d^e dp dt . 



y dv drj dri dv ' 



The two last expressions for the value of ^-r-Yv 7; indicate that the 

 value of y^ „ in different parts of the diagram will be indicated pro- 

 portionally by the segments into which vertical lines are divided by a 

 system of equidifferent isopiestics, and also by the segments into 

 which horizontal lines are divided by a system of equidifferent iso- 

 thermals. These results might also be derived directly from the 

 propositions on page 313. 



As, in almost all cases, the pressure of a body is increased when it 



dp 



receives heat without change of volume, ~ is in general positive, and 



the same will be true of y^, under the assumptions which we have 

 made in regard to the directions of the axes (page 330) and the defini- 

 tion of a positive area (page 331). 



In the estimation of work and heat it may often be of use to con- 

 sider the deformation necessary to reduce the diagram to one of 

 constant scale for work and heat. Now if the diagram be so deformed. 



