80 TREATISE ON ALTERNATING CURRENTS. 



and by Ohm's law 

 therefore 

 therefore 



where T is the periodic time. 

 Therefore 



2RP = F 2 - Fi 2 - F 2 2 

 or 



F* - F! 2 - F 2 2 



!>// 



In proving this formula (which gives the three-voltmeter 

 method of measuring the power given to an inductive circuit), no 

 assumption whatever is made respecting the shape of the E.M.F. 

 curves. 



Now, let /be the E.M.S. current flowing through ABC; then, 

 since E is non-inductive, we have 



therefore 



Y* 

 P = I. 



F 2 Fi 2 F 2 2 



P - I - 



2Fk 



If we now put 



F 2 - F, 2 - F, 2 



s 



which is admissible, since v is never greater than v\ + va> we get 



P = F 2 / cos 



which is the same as if the quantities were simple sine functions of 

 the time, and the angle of lag or lead. 



The point to notice is that the expression for cos is what it 

 would be if the P.D.s were definite vectors. This is seen by 

 reference to Fig. 31. 



We may call the equivalent phase-difference between 



