THE POLYPHASE CIRCUIT. 311 



Let e A , e b , e ( , = the voltages of the branch circuits 



respectively. 



c iv , c b , c c = the currents in the same circuits. 

 e A , e B , e c = the voltages between the lines. 

 C A , C B , c c = the currents in the line conductors. 



Also consider the directions indicated by the arrows in 

 the figure to be positive directions, in order to give 

 definite meanings to the signs to be employed. 

 The power to be measured 



= W = e & c a + ? b c b + e, c c (1) 



At the given instant it is evident from an inspection of 

 the figure that 



* a = + e c (2) 



also c,. = c a + c, (3) 



and C A = c,. - c b (4) 



The reading of wattmeter W l at any moment 



= W ! = A c ( . - e a (c ft + c,,) - e n c a + e. A c h 

 Substituting from equation (2)i 



JPi = . c a + Ct, (e b + e c ) = e a c a + e,, c,, + ,. c h . (5) 

 Similarly the reading of wattmeter W. 2 



= W., = ,. C A = e,. (c r c b ) - e t . c,. e,, c h . . . (6) 

 Adding equations (5) and (6) 



W, + W. 2 = e & c a + e b c,, + e,. c,. > (7) 



Since the wattmeters in each case read the mean of 

 the products of current and volts, the sum of the readings 

 will be the mean value of the watts W as given in 

 equation (7). 



The proof just given is based on the distribution of 

 currents and voltages in a mesh-connected circuit. A 

 similar proof may be obtained for a star-connected 

 circuit. This is, however, hardly necessary, since from 

 the method of connecting the wattmeters, it is evident that 

 they will read the power transmitted by the line quita 

 independently of the manner in which the branch circuits 

 may be connected. Thus the circuits a, 6, c, in Fig. 156, 

 might equally well be connected in star, instead of mesh, 

 and the sum of the readings of the two wattmeters 

 would still give the total power passing in the lines. 



In the present case, as in the Method II. of measure- 

 ment by one wattmeter given above, it is to be noticed 



