Self-induction of Wires. 183 



altogether, we see by (18c) that the differential equation of C 6 is 

 f z + (Z 1 + Z,)(Z, + Z t ) l 



e ~ \ z ° + z 1+ z 2 +z 3 +z 4 / 0e ' 



manipulating the Z's like resistances. The absence of 

 branch 5 thus reduces the number of free-subsidence systems 

 to two. Now, if we choose x x = x 2 , we shall make 



(L 1 +L,)/(B 1 + B,) = (L 2 + L 4 )/(R 2 + E 4 ), 

 or the time-constants of the two branches 1 + 3 and 2+4 

 equal. Then one of the p's is 



and this is only concerned in the free subsidence of current 

 in the circuit AB 1 CB 2 A. Consequently the second p, which is 



__ (Ri + R 3 )^2 + Reffii + R2) 



£2 (L 1 + L 3 )R 2 + L 6 (R 1 + R 2 ) ? 



is alone concerned in the setting-up of current by the im- 

 pressed force in 6 ; and the current divides between AB X C 

 and AB 2 C in the ratio of their conductances, in the variable 

 period as well as finally. In fact, the fraction in the above 

 equation of 6 will be found to contain Z x + Z 3 as a factor in 

 its numerator and denominator, thus excluding the p± root, 

 so far as e is concerned. On the other hand, if we choose 

 x 1 = x s , we do not have equality of time-constants of AB X C 

 and AB 2 C, so that there are two _p's concerned, which are not 

 those given ; and the current C 6 does not, in the variable 

 period, divide between AB X C and AB 2 C in the ratio of their 

 conductances, but only finally. 



In the above statement it was assumed that when Lj and 

 L 2 were chosen, it was not so as to make x x = x 2 . When this 

 happens, however, it is only the ratio of L 3 to L 4 that becomes 

 fixed, for we have x^=x 4: = anything. 



Similarly, when Lj and L 3 are so chosen that Xi=x s , we 

 shall have x 2 = x± = anything, so that only the ratio of L 2 to 

 L 4 is fixed. 



And if L 3 , L 4 be so chosen that x s = x 4j then x ± = x 2 = any- 

 thing, only fixing the ratio of L x to L 2 . But should x B not 

 = x 4 , then we require x 1 = x 3 and x 2 = x 4 , thus fixing L^ and L 2 . 



And if L 2 , L 4 be so chosen that x 2 = x A , then xx = x 3 = any- 

 thing, only fixing the ratio of R x to R 3 . But if so that x 2 not 

 ■~x 4 , then x 1 — x 2 and # 3 = # 4 fix L x and L 3 . 



There are yet two other pairs that may be initially chosen, 

 and with somewhat different results. Let it be L x and L 4 that 

 are chosen ; if not so as to make x 1 =x 4: , there are two ways 



