Self-induction of Wires. 71 



which makes C 5 always zero if e l = e 2 , e z — e^ and Z 1 7j 4l = Z 2 Z 3 . 

 As an example, let <? 2 = 0, e 4 =0 ; then, if there is conjugacy 

 of 5 and 6, and also 



e 1 /(Z 1 + Z 2 ) = e 3 /(Z B + Z,), .... (20d) 



the impressed forces are also balanced. Putting, therefore, 

 batteries in sides 1 and 3, and letting them work an inter- 

 mitter in branch 6, we obtain a simultaneous balance of their 

 resistances and E.M.F/s, and know the ratio of the latter. 

 If self-induction be negligible, we may take Z as R, the 

 resistance ; if not negligible, it must be separately balanced. 

 But should there be mutual induction between different 

 branches, this working-out of problems relating to transient 

 states by merely turning R to Z partly fails. We may then 

 proceed thus: — As before, write down the equations of E.M.F. 

 in the circuits ABjB 2 A and CB^C, using the six independent 

 inductances of these and of the circuit CAB 2 C. Thus, 



e\ = RiCj + R 5 C 5 - R 2 C 2 +p(m 1 C 1 + ra 13 C 3 + ?n 16 C 6 ) , ) .^ifa 



a=R I c t -B 4 c 4 -B B q 6 +^(fii il o 1 +«» I c i +m w o e ),j ' ; 



if there is an impressed force in side 1. As before, eliminate 

 C 2 , C 3 , and C 4 by (16d), and we obtain 



« l + (Rj-pw 16 )C 6 = {R ] +R 2 ^(m 1 +m 13 )}C 1 + (R 5 -);%)C 5 , j 



(R 4 -pm 36 )C 6 = { R 3 + R 4 +p (m 3 + m u ) } Cj - (R 3 + R 4 + R 5 +pm 3 ) C 5 , J 



which, by solution for C 5 , give its differential equation at once 

 in terms of e± and C 6 . To be independent of C 6 , we require 



(R 2 —pm 16 ) { R 3 4- R 4 +p {m 3 + ??? 13 ) } 



= (R 4 -p?7 36 ){R 1 + R 2 + i? (^ 1 + ^i3)}, • (23d) 

 which, expanded, gives us the three equations (§d) again, 

 showing that C 5 depends upon e x and the nature of sides 1, 2, 

 3, and 4, subject to (2&Z), and of 5, but is independent of the 

 nature of 6 altogether, except in the fact that the mutual 

 induction between branch 6 and other parts of the system 

 must be of the proper amounts to satisfy (23(/) or (6d). 



The extension that is naturally suggested of this property 

 to any network whose branches may be complex, and not 

 independent, is briefly as follows. The equations of E.M.F. 

 of the branches will be of the form 



« l + V 1 = ZA + Z 1 ^ a + Z 18 C, + ... ; -] 



^ 2 + V 2 = Z 21 C 1 + Z2C 2 + Z 23 C 3 + ...,[ . . . (24tf) 



wherein the Z's are differentiation-operators. 



