312 ,/. W. Glhbs on Graphical Methods in the 



The value of these quantities may be calculated from equations (2) 

 and (3), 



dH^ tdi], 

 i. e., W=Jpdi; (5) 



// =ftdr], (6) 



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

 ])ath. If the direction of the ])ath 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 circuity and the 

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

 (1), which when integrated for this case becomes z=.II— W. 



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

 })0sitive or negative according to the direction of the circuit which 

 circumscribes it. The direction in which areas must be circum- 

 scribed in order that their value may be positive, is of course arbi- 

 trary. In other words, if x and i/ are the rectangular co-ordinates, 

 we may define an area either us fi/dx, or asfxdy. 



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 tlie work done in each of the lines whicli 

 separate the partial areas appears twice and with contrary signs in 

 the sxim 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 tlie 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 evi- 

 dent from the consideration that a reversal of this direction simply 

 changes the sign of both terms of the ratio. To prove that the ratio 



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

 tions 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. 



