4 



GRAPHICAL METHODS IN THE 



the integration being carried on from the beginning to the end of the 

 path. If the direction of the path is reversed, W and H change their 

 signs, remaining the same in absolute value. 



If the changes of state of the body form a cycle, i.e., if the final 

 state is the same as the initial, the path becomes a circuit, and the 

 work done and heat received are equal, as may be seen from equation 

 (1), which when integrated for this case becomes = H W. 



The circuit will enclose a certain area, which we may consider as 

 positive or negative according to the direction of the circuit which 

 circumscribes it. The direction in which areas must be circumscribed 

 in order that their value may be positive, is of course arbitrary. In 

 other words, if x and y are the rectangular co-ordinates, we may 

 define an area either a.sj'ydx, or &sjxdy. 



If an area be divided into any number of parts, the work done in 

 the circuit bounding the whole area is equal to the sum of the work 

 done in all the circuits bounding the partial areas. This is evident 

 from the consideration, that the work done in each of the lines which 

 separate the partial areas appears twice and with contrary signs in 

 the sum of the work done in the circuits bounding the partial areas. 

 Also the heat received in the circuit bounding the whole area is equal 

 to the sum of the heat received in all the circuits bounding the 

 partial areas.* 



If all the dimensions of a circuit are infinitely small, the ratio of. 

 the included area to the work or heat of the circuit is independent of 



the shape of the circuit and the 

 direction in which it is described, 

 and varies only with its position 

 in the diagram. That this ratio 

 is independent of the direction in 

 which the circuit is described, is 

 evident from the consideration 

 that a reversal of this direction 

 simply changes the sign of both 

 terms of the ratio. To prove that 

 the ratio is independent of the 

 shape of the circuit, let us suppose 

 Fig> L the area ABODE (fig. 1) divided 



up by an infinite number of isometrics v^o v V 2 v 2 , etc., with equal 

 differences of volume dv, and an infinite number of isopiestics p l p l , 

 P 2 p 2 , etc., with equal differences of pressure dp. Now from the 



* The conception of areas as positive or negative renders it unnecessary in propositions 

 of this kind to state explicitly the direction in which the circuits are to be described. 

 For the directions of the circuits are determined by the signs of the areas, and the signs 

 of the partial areas must be the same as that of the area out of which they were formed. 



