Self-induction of Wires. 181 



or other indicator in 5. Mutual inductances will be denoted 

 by M ; thus, M 12 Oj is the electromotive impulse in 2 due to 

 the stoppage of the current Q x in 1 ; similarly M 12 2 is the 

 impulse in 1 due to stopping 2 . 



Deferring mutual induction for the present, though not 

 confining self-induction to be of the electromagnetic kind 

 only, but to include electrostatic if required, the condition of 

 conjugacy is that the potentials at Bj and B 2 be always 

 equal. Therefore 



V 1= =V 2 , and V 3 = V 4 ; .... (20c) 



so, if V = ZC, 



ZA=Z 2 C 2 , and Z 3 C 3 =Z 4 C 4 . . . . (21c) 



But, by continuity, C 1 = C 3 , and C 2 = C 4 at every moment 

 (including equality of all their differential coefficients) ; so 

 that (21c) becomes 



Z 1 Ci = Z 2 C 2 , Z 3 \J 1 = Z 4 C 2 ; .... (22c) 



consequently 



Z 1 Z±-Z 2 Z 3 = 0=f (23c) 



is the complex condition of conjugacy. This function is the 

 / of the previous investigation. 



When the self-induction is of the electromagnetic kind, 

 Z = R + Ljt? ; so that, arranging / in powers of p, 



0= (R^ 4 -R 2 R 3 ) + (R 1 L 4 + R 4 L l -R 2 L 3 -R 3 L 2 ) i? + (LjL 4 - L 2 L 3 )p 2 . 



Therefore, if ^j = L/R, the time-constant of a branch, we have 

 three conditions to satisfy, namely, 



R 1 R 4 =R 2 R 3 , I (25c) 



« 1 +a? 4 = ^? 2 + ^3, I (26c) 



L 1 L 4 =L 2 L 3 . I (27c) 



" If the first condition is fulfilled, there will be no final 

 current in 5 when a steady impressed force is put in 6. This 

 is the condition for a true resistance balance. 



ie If, in addition to this, the second condition is also satis- 

 fied, the integral extra current in 5 on making or breaking 6 

 is zero, besides the steady current being zero, (25c) and (26c) 

 together therefore give an approximate induction balance 

 with a true resistance balance. 



" If, in addition to (25c) and (2Qc), the third condition is 

 satisfied, the extra current is zero at every moment during 

 the transient state, and the balance is exact however the im- 

 pressed force in 6 vary. 



