and Conductance Operators. 501 



respectively, at time t after insertion of the condenser, and due 

 entirely to its initial charge. Equations (46) above express 

 them, if u and w have the special ratio proper at the condenser, 



given by U7=-SpM, (48) 



because the current equals the rate of decrease of its charge. 

 Initially, we have e = %Au and 2Aw = 0. So, making use of 

 the conjugate property"*, we have 



S*m = 2(U / ,-T p )A, (49) 



if V p be the electric and T p the magnetic energy in the normal 

 system. But the following property of the resistance-operator 

 is also true *, 77 



T p -U„=^ ; (50) 



that is, dZJdp is the impulsive inductance in the p system at 

 a place where the resistance-operator is Z ly p being a root of 

 Zi = ; just as dZ 1 /dp withp = is the impulsive inductance 

 (complete) at the same place. Using (50) in (49) gives 



A=-(S,u)-(t C sfO ( 51 ) 



Now use (48) in (51) and insert the resulting A in the second 

 of (46), and there results 



c = 2 p^> ( 52 > 



where the accent means differentiation to p. This is the com- 

 plete subsidence solution. Now increase S infinitely, keeping 

 e constant. Z l ultimately becomes Z ; but, in doing so, one 

 root of Z x = becomes zero. We have, by (47), and remem- 

 bering that Z x = 0, 



p Z/=-(Sj9)- 1 +pZ / =Z+ 7? Z'; . . . (53) 



so, when S = co and Z = 0, we have pZi'=pZ' for all roots ex- 

 cept the one just mentioned, in which case p tends to zero and 

 Z' is finite, making in the limit pZi=Z 0j by (53), where Z is 

 the p = value of Z, or the steady resistance. Therefore, 



fi " ally ' C =i + 2 P >'' ^ 



where the summation extends over the roots of Z = 0, shows 

 the manner of establishment of the current by the impressed 

 force e. The use of this equation (54), even in comparatively 

 elementary problems, leads to a considerable saving of labour, 

 whilst in cases involving partial differential equations it is 

 invaluable"*. To extend it to show the rise of the current at 

 * « On the Self-induction of Wires," Phil. Mag. Oct. 1886, 



