THE USE OF COMPLEX QUANTITY. 



89 



and / are stationary with respect to the reference axis. Let t\ and i n be the 

 components of /, and let e l and e n be the components of J5, then we may write : 



and 



in which the letter j is merely a mark used to distinguish the /-component, and the 

 plus sign means no more than to say, for example, that / consists of an ^--component 

 ?! and a /-component i n . 



The utility of equations (z) and (it] can beat once exemplified by considering the 

 composition and resolution (addition and subtraction) of currents or of electromotive 

 forces. The algebraic meaning of the index letter j will be completely determined 

 when we consider the idea of impedance as an algebraic operator. 



Example i. Addition. Given three harmonic alternating electromotive forces, 

 of the same frequency, which are to be added together. If these three electromotive 

 forces be represented by three lines A, B, C, in a clock diagram, then their sum 

 will be represented by the line which is the vector sum or resultant of A, B and C ; 

 the Jtr-component of this resultant will be equal to the sum of the .r-components of 

 A, B and C ; and its ^-component will be equal to the sum of the /-components of 

 A, B and C. That is to say, electromotive forces (or currents) are added by adding 

 their ^--components to give the jr-component of the sum, and by adding their /-com- 

 ponents to give the /-component of the sum. Thus (^ -f^u) + (/i ~\rJf\\) 



Example 2. Subtraction. Given two harmonic electromotive forces, of the same 

 frequency, one of which is to be sub- 

 tracted from the other. If these elec- 

 tromotive forces be represented by lines 

 A and B in a clock diagram, then their 

 difference will be represented by the 

 line which is the vector difference of A 

 and B, the .r-component of this line 

 will be the difference of the ^-compo- 

 nents of A and B, and its /-component 

 will be the difference of the /-compo- g> 



nents of A and B. That is to say, 



electromotive forces (or currents) are subtracted by subtracting their ^-compo- 

 nents to give the jc-component of their difference, and by subtracting their /-compo- 

 nents to give the /-component of their difference. Thus (<i'4"/*u) (fi^Jfn) 



(b*) The idea of impedance as an algebraic operator. Let the current vector /, 

 Fig. 84, be chosen as the reference axis, then the Jt-component ( parallel to / ) of E is 

 equal to RI, and the /-component (90 ahead of 7) of E is XL Therefore, 

 marking the /-component with the index letter j, we have 





05) 



