210 Mr. O. Heaviside on the 



resistance and inductance, so that in (96c) and (97c) we have 

 Rj=R 4 and Li=L 4 , the first reduces to 



0=M 14 -M 23 , (99c) 



so that it is first of all absolutely necessary that M 14 =M 23 , if 

 the balance is to be preserved; whilst, subject to this, the 

 second condition reduces to 



0=(M 24 -M 13 )(M 34 -M 12 ), . . . (100c) 



so that either M 24 =M 13 , or else M 34 =M 12 . Thus there are 

 two ways of preserving the balance when all four branches 

 are equal, viz. M 14 =M 23 and M 24 =M 13 , independent of the 

 values of M 12 and M 34 ; and M 14 =M 23 and M 34 =M 12 , inde- 

 pendent of the values of M 24 and M 13 . 



The verification of these properties, (98c) and later, makes 

 some very pretty experiments, especially when the four 

 branches consist, not merely of one coil each, but of two or 

 more. The meanings of some of the simpler balances are 

 easily reasoned out without mathematical examination of the 

 theory ; but this is not the case when there is simultaneous 

 induction between many coils, and their resultant action on 

 the telephone-branch is required. 



Returning to (96c) and (97c), the nearest approach we can 

 possibly make to independence of the self-induction balance 

 of the values of all the M's therein concerned, consistent with 

 keeping wires 3 and 4 away from one another for experi- 

 mental purposes, is by winding the equal wires 1 and 2 

 together. Then, whether they be joined up straight, which 

 makes M 13 = M 23 and M 14 =M 24 identically, or reversed, making 

 M 13 = — M 23 and M 14 = — M 24 , we shall find that 



M 14 =M 23 

 is the necessary and sufficient condition of preservation of 

 balance. 



At first sight it looks as if M 31 and M 32 must cancel one 

 another when wires 1 and 2 are reversed. But although 1 

 and 2 cancel on 3, yet 3 does not cancel on 1 and 2 as regards 

 the telephone in 5. The effects are added. On the other 

 hand, when wires 1 and 2 are straight, 3 cancels on them as 

 regards the telephone, but 1 and 2 add their effects on 3. 

 Similar remarks apply to the action between 4 and the equal 

 wires 1 and 2 when straight or reversed ; hence the necessity 

 of the condition represented by the last equation. 



On the other hand, M 61 and M 62 cancel when 1 and 2 are 

 straight, and add their effects when they are reversed ; whilst 

 M 51 and M 52 cancel when 1 and 2 are reversed, and add their 

 effects when they are straight, results which are immediately 



