L 3 -L 4 =M 24 -M ia + M 13 ^-M 24 f 3 , 



I 



Self-induction of Wires. 209 



If we compare the two general conditions (83c) , (84c) , we 

 shall see that whenever 



we may obtain the reduced forms of the conditions by adding 

 together the values of L 3 — L 4 given by every one of the M's 

 concerned. We may therefore bracket together certain sets 

 of the M's. To illustrate this, suppose that M 13 and M 24 are 

 existent together, and all the other M's are zero. Then (92c) 

 and (93c) give, by addition, 



L 3 -L 4 =(M 24 -M 13 )(l-g), 



l 4_m h 

 13 l; iVl24 v 



which are the conditions required. 



Similarly M 12 and M^ may be bracketed. Also M 61 , M 62 , 

 M 63 , M 64 , and M 65 . Also M 61 , M 52 , M^, M 54 , and M a . But 

 M 14 and M 23 will not bracket. 



As already observed, the self-induction balance (28c) (29c) 

 is independent of M 12 and M 34 , when these are the sole mutual 

 inductances concerned ; that is, when Rj = R 2 , I^i == L 2 , H 3 = R 4 , 

 L 3 =L 4 . By (92c) and (93c) we see that independence of 

 M 13 and M 24 is secured by making all four branches 1, 2, 3, 4 

 equal in resistance and inductance. 



But it is unsafe to draw conclusions relating to inde- 

 pendence when several coils mutually influence, from the 

 conditions securing balance when only one pair of coils at a 

 time influence one another. Let us examine what (83 c) and 

 (84 c) reduce to when there is induction between all the four 

 branches 1, 2, 3, 4, but none between 5 and the rest or 

 between 6 and the rest. Put all M's = which have either 

 5 or 6 in their double suffixes, and put L 4 — L 3 . Then we may 

 write the conditions thus :— 



0=(l + R 4 /R 1 )(Mi4-M 23 ) + (l-R 4 /R 1 )(M 24 -M 13 ), (96c 



0=(L ] + L 4 )(M 14 -M 23 ) +(L 1 -L 4 )(M 24 -M 13 ) + Mf 3 -M? 4 



+ (M 24 -M 13 )(M 34 -M 12 )+ (M 14 -M 23 )(M 24 + M 13 -M 12 -M 34 ), (97c 



The simplest way of satisfying these is by making 



M 14 =M 23 and M 24 =M 13 (98c) 



If these equalities be satisfied, we have independence of M 12 

 and M 34 . 



Now, if we make the four branches 1, 2, 3, 4 equal in 



