Self-induction of Wires. 349 



To find A as a function of z u we might, since 2 Ku is the 

 V set up by unit e at z 1} expand this state by the former 

 process of integration. But the following method, though 

 unnecessary for the present purpose, has the advantage of 

 being applicable to cases in which VC is not zero at the ter- 

 minals, but V = ZC instead. It is clear that the integration 

 process, including the energy in the terminal apparatus, would 

 be very lengthy, and would require a detailed knowledge of 

 the terminal combinations. This is avoided by replacing the 

 impressed force at z Y by a charged condenser ; when, clearly, 

 the integration is confined to one spot. Let S x be the capacity 

 and V the difference of potential of a condenser inserted at z^. 

 If we increase S x infinitely it becomes mathematically equiva- 

 lent to an impressed force V , without the condenser. 



Suppose HAiv'e pt is the current at z at time t after the intro- 



duction of the condenser, of finite capacity ; then, since — SiV 

 is the current leaving the condenser, or the current at z ly we 

 have 



-S 1 V=2A M ? 1 '€P', 



wj being the value of iv 1 at z x . The expansion of V is there- 

 fore 



V =-SAV/Sii>, 



initially ; and the mutual potential energy of the initial charge 

 of the condenser and of Ihe normal u' corresponding to w r 

 must be 



s 1 v (-«> 1 7S li >)=-y v/i?. 



But since there is, initially, electric energy only at z v and 

 magnetic energy nowhere at all, the only term in the nume- 

 rator of A will be that due to the condenser, or this — Yoiv^/p; 

 hence 



A=-Y wJpM, 



where M is the 2(U— T) of the complete normal system, as 

 modified by the presence of the condenser, is the value of A 

 in Y = *2Au'e pt , making 



expressing the effect at time t after the introduction of the 

 condenser, and due to its initial charge. 



So far Si has been finite, and consequently u f , w f , M, and 

 p depend on its capacity as well as on the line and terminal 

 conditions. But on infinitely increasing its capacity, v! and 

 w' become u and w, the same as if the condenser were non- 

 existent. Therefore 



Y=-ZY (w 1 /pM)ueP t .... (134) 



