Forced Vibrations of Electromagnetic Systems. 149 



C now mounts up infinitely. But the leakage-current, which 

 is KV, becomes steady, as (76) shows. 



In connexion with this subject I should remark that the 

 non-distortional circuit produced by taking R/L = K/S is of 

 immense assistance, as its properties can be investigated in 

 full detail by elementary methods, and are most instructive in 

 respect to the distortional circuits in question above *. 



13. Modifications made by Terminal Apparatus. Certain 

 cases easily brought to full realization. — Suppose that the ter- 

 minal conditions in the preceding are V = — Z C and V=Z 1 C, 

 Z and Z x being the "resistance operators " of terminal appa- 

 ratus at A and B respectively. In a certain class of cases the 

 determinantal equation so simplifies as to render full realiza- 

 tion possible in an elementary manner. Thus, the resistance- 

 operator of the circuit, reckoned at A, is f 



& _7 , (R + ^p)l(tanml)/ml + Z 1 ( . 



9 ~ ° + 1 4- (K + Sp)/Z 1 (tan ml)JmV ' K J 



m 2 =-(R + Lp)(K + Hp) (79) 



where 



That is, e = <f)C is the linear differential equation of the current 

 at A. Now, to illustrate the reductions obviously possible, 

 let Z =0, and 



Z 1 = n 1 Z(R + L i ?) (80) 



This makes the apparatus at B a coil whose time-constant is 

 L/R, and reduces <£ to 



♦-CB + i*)*f=r + -,){i-^^}" , .-..(<«) 



so that the roots of <p = are given by 



U + Lp = (82) 



tan ml-\-mln i = 0; (83) 



i. e. a solitary root p= — B/L and the roots of (83), which is 

 an elementary well-known form of determinantal equation. 



The complete solution due to the insertion of the steady im- 

 pressed force e at A will be given byf 



* "Electromagnetic Induction and its Propagation," Arts. xl. to 1. 

 f " On the Self-induction of Wires," Part IV. 



X lb. Parts III. and IV. Phil. Mag. Oct. and Nov. 1886 ; or « On Re- 

 sistance and Conductance Operators," Phil. Mag. Dec. 1887, § 17, p. 500. 



