Power given by any Electric Current to any Circuit. 435 



where e is a small fraction, i.., let the errors be each the same small 

 raction of the correct value, then the probable value of 



4 (FVF 2 + V?#V?+ F 2 2 e 2 F 2 2 ), 



d_ , 

 ._ 4e2 



s 



Ull 



* 



(7 2 -F 1 2 -F 2 ) 2 

 Let the non-inductive resistance have such a value that 



F, = Fi (8), 



2 being already defined, the square root of the mean square of the 

 .D. between its terminals, and V L the square root of the mean square 

 the P.D. between the terminals of the circuit the power given to 

 which we desire to measure. Then we wish to find the value of x 

 that will make eZTF/TFa minimum. 



Let be the angle of lag between the current in the circuit ac and 

 .e P.D. at the terminals of ab (fig. ].), then is the angle of lag 

 between the P.D. at the terminals of ab and the P.D. at the terminals 

 be. Hence, since 



i, and v z being the instantaneous values of the P.Ds., 



F 2 = F 1 2 +F 2 2 +2F 1 F 2 cos0 (9). 



Eliminating F, FI, and F 2 by means of equations (7), (8), and 

 )), we have 



/dW\ z _ 2 (l+a 2 +2a;cos0) 2 +l + a; 4 



\W) ~ 435 2 COS 2 



Now cos depends on the circuit, the power given to which we 

 sire to measure, and is independent of x. Hence differentiating 

 nth respect to y and equating to nought in the usual manner, we 



id that x equal to unity makes a minimum. 



W 



Hence, inaccuracies in the three readings of the voltmeter, or in 

 graduation of its, scale, produce the least effect in this method of 

 sasuring power when the P.D. between the terminals of the non- 

 iductive resistance is equal to the P.D. at the terminals of the circuit 

 under test. 



The next point to consider is, what is the percentage error made in 

 leasuring the power by this method compared with the percentage 



made in reading one of the P.Ds. 

 Let x equal unity, then 



