Self-induction of Wires. 207 



the two conditions in each case besides the resistance-balance 

 condition, which is always the same. 

 All M's=0, except M 36 . 



RjK^j + x±— -x 2 — # 3 ) = (R T + R 2 ) M 36 , 



-la-^ + wmJ) * ' (75c) 



AllM's = 0, except M 46 . 



R 1 R 4 (# 1 +# 4 — #2— ^3) = -(K 1 + E 2 )M 46 ,l . . 



L 1 L 4 -L 2 L 3 =-(L 1 + L 4 )M 46 .J ' ' ^ C; 



As these only differ in the sign of the M, we may unite 

 these two cases, allowing induction between 6 and 3, and 6 

 and 4. The two conditions will be got by writing M 38 — M 46 

 for M 36 in (75c). 



All M's = 0, except M 56 (Prof. Hughes's method). 



= R 1 R 4 (.£ 1 +^ 4 -^ 2 -tf 3 )+M 56 (R 1 + R 2 + R 3 + R 4 ), 



0=L 1 L 4 -L 2 L 3 + M 56 (L 1 + L 2 + L 3 + L 4 ). 



Now choose a ratio of equality, R 1 = R 2 , 1^ = 1^, which is 

 the really practical way of using induction-balances in 

 general. In the M 36 case the two conditions (75 c) unite to 

 form the single condition 



L 4 -L 3 =2M 36 , (78c) 



and in the M 46 case (76c) unite to form the single condition 



L 4 -L 3 =-2M 46 (79c) 



We know already that the same occurs in the simple Bridge 

 (29c), making 



L 4 =L 3 ; (80c) 



so that we have three ways of uniting the second and third 

 conditions. Now examine all the other M's, one at a time, 

 on the same assumption, R 1 = R 2 , L^Lg. With M 12 we 

 obtain 



(L 4 -L 3 )(L 1 -M 12 )=0, and L 4 =L 3 . 



But Lj— M 12 cannot vanish ; so that 



L 4 =L 3 (81c) 



is the single condition. Similarly, in case of M 43 , 



L 4 =L 3 (82c) 



again. All these, (77 c) to (82 c), were given in the paper 

 referred to ; the last two mean that M 12 and M 34 have abso- 

 lutely no influence on the balance of self-induction. 



All the rest are double conditions. Thus, in A] and A 2 



