ELr.CTRIC CIRCriT '/lir.ORV 369 



operator d/dl. By aid of the fundanu'nlal lonniila lliese eciualioiis 

 may be written down as the following sinuillaneous integral efjuations: 



hit) =^£dyAu{t-y) {E{y) + M^-^^[J(y)h{y)]) 

 h{t) = M^£dyA ,,{t - y) ~ [fiy) I,{y)] . 



In these equations, An(t) and Ai-iiO denote the indicia! admittances 

 of the primary and secondary circuits respectively (when M = 0): 

 that is, the currents in these circuits in response to a unit e.m.f. (zero 

 before, unity after time / = 0). We assume, of course, that they are 

 known or can be determined by usual methods. 



It follows at once that the formal solution of these equations is 

 the infinite series: 



h{t)=Xo{t)+X,(t)+X,(t)^ . . . +X2„(/)+ . . . 



/2(/) = Fl(/)+F3(0+F5(0+ . .. 



in which the successive terms of the series are defined as follows: 



Xoit) =-^j£dyAn{t-y)E{y) =Io{t), 



Y,{t)=Mj^-£dyA,,{t-y) ^[f{y)Xo{y)h 

 X,{t)=M^£dyA,,{t-y)f^\f{y) F,(j)], 



Y,{t) =M^£dyA,2{t-y)^[fiy)X,(y)], etc. 

 In the light of formula 



m=^£f(y)A(t-y)dy 



the physical interpretation of the series solutions follows at once : 

 Thus, Xo{t) is equal to the current Io{t) flowing in the isolated primary 

 in response to the applied e.m.f. £(/) ; the first component current 

 Yi(t) in the secondary is equal to the current which would flow in the 

 isolated secondary in response to the applied e.m.f. M(d/dt)f(t)Xo{t); 

 Xi^t), the second component current in the primary, is equal to the cur- 

 rent in the isolated primary in response to the applied e.m.f. M{d/dt) 



