478 THE ELECTROMAGNETIC FIELD. [PT. III. CH. XII. 



the equation with the sine on the right. In this way by separa- 

 tion of the real and the imaginary we are enabled to use the 

 exponential function, which retains its form on differentiation, 

 while the sine and cosine interchange. Accordingly writing the 

 equation 



(2) L d -jj- t +RI=E^\ 



we may get a particular solution by assuming /= Ae i<at , inserting 

 which in the equation gives, on removing the factor e ivtt , 



(3) (Lito + R) A = E . 

 This determines the complex constant A as 



E, ^ 

 Lico + R 



so that the solution of the equation (2) is 



A t _E Q (R Lico) (cos cot + i sin cat) 

 LW + R* 



Taking the real part we obtain for the solution of the equation (i), 



j._E (R cos cot + Leo sin cot) 

 + R* 



This assumes a more convenient form if we determine two 

 constants a and J so that 



R cos a. Leo sin a 



gvng 



(4) tan a = ~ , J 

 when the solution becomes 



(5) 7 = 4 cos (art -a). 



J 



We may obtain this result, and at the same time graphically 

 represent the relations of the current and electromotive-force by 

 making use of the fundamental properties of complex quantities. 

 The complex quantity EtfF* 1 has the modulus E and the argument 

 cot, and is therefore represented by a vector of length E Q making 

 an angle cot with the real axis, that is a vector revolving about the 

 origin with angular velocity co. The projection of this vector on 

 the real axis represents the impressed electromotive-force in the 



