GRAPHICAL METHODS. 43 



angle aOQ = p (t t'), and if QM is drawn at right angles to Oa } we 

 have 



QM = OQsinp(t - t') 



= r sin (pt - 0) 

 where 



pf = = angle POQ 



If we take the sum of PN and QM, we get 

 PN + QM=TL 



where T is determined by completing the parallelogram having 

 OP and OQ as adjacent sides, and L is the foot of the perpendicular 

 from T on Oa produced. 



Thus the sum of PN and QM is a sine function of the time 

 having the same periodic time as either PN or QM, but differing 

 in phase from either PN or QM, and having its maximum value 

 given by the diagonal through of the parallelogram POQT. 



Hence, if the maximum values of two alternating currents, or 

 E.M.F.'s were represented by OP and OQ respectively, the maximum 

 value of the current, or E.M.F. resulting from the two when 

 superimposed, is determined by the law of composition of Vectors, 

 and is the diagonal through of the parallelogram with OP and 

 OQ as adjacent sides. 



The corresponding instantaneous values are given by the 

 ordinates PN, QM, and TL respectively, as the parallelogram POQT 

 is rotated round with uniform angular velocity. 



31. Suppose we consider the case of a single inductive circuit, 

 whose self-induction is L and resistance r. Let E be the maximum 

 value of the P.D. impressed between 

 the terminals of the circuit and / 

 the maximum value of the current 

 produced. 



Let OA (Fig. 10) represent in 

 magnitude rl, that is the E.M.F. 

 necessary to drive the maximum 

 current against the resistance. This 

 is in phase with the current. The 

 maximum value of the E.M.F. due 

 to self-induction is pLI (see 21, 



Chap. III.), and is represented by OL drawn at right angles to OA 

 and 90 behind it. The maximum value of the E.M.F. necessary 



