Self-induction of Wires, 175 



AB 2 C each consist of a number of coils in sequence, they will 

 balance if the coils are alike, each for each, in the two lines, 

 and are similarly placed with respect to one another. But 

 the lines will easily balance under simpler conditions, coeffi- 

 cients of self-induction being additive, like resistances ; and 

 it is only necessary that the total self-inductions of AB X and 

 AB 2 (including mutual induction of their parts) be equal, and 

 likewise of B^ and B 2 0. Again, if a coil a x in the branch 

 ABj have another coil b± in its neighbourhood (not in either 

 line, but independent), and a 2 be a copy of a ly in the branch 

 AB 2 , we can complete the balance by placing a coil b 2 which 

 is a copy of b\ in the neighbourhood of the coil a 2J so that the 

 action between a 1 and b± is the same as that between a 2 and b 2 . 

 But it is not necessary for bi and b 2 to be copies of one another 

 except in the two particulars of resistance and self-induction; 

 whilst as regards their positions with respect to % and a 2 , we 

 only require the mutual induction of a x and b Y to equal that of 

 a 2 and b 2 . 



On the other hand, if b x be a piece of metal, not a coil of 

 fine wire, that is placed near the coil a 1} many more specifica- 

 tions are required to make a copy of it. The piece of metal 

 is not a linear conductor ; and, although no doubt only a small 

 number (instead of an infinite number) of degrees of freedom 

 allowed for would be sufficient to make a practical balance, 

 yet, as we have not the means of simply analyzing pieces of 

 metal (like coils) into a few distinct elements, we must generally 

 make a copy of b x by means of a similar piece of the same 

 metal, b 2 , and place it with respect to a 2 as b x is to a 1} to secure 

 a good balance. But very near balances may be sometimes 

 obtained by using quite dissimilar pieces of metal, dissimilarly 

 placed. 



So far, copy signifies equality in certain properties. But 

 one line need be merely a reduced copy of the other. It is 

 only when we inquire into what makes one line a reduced copy 

 of another, that we require to examine fully the mathematical 

 conditions of the case in question. In the state of steady flow 

 the matter is simple enough. If AB X has n times the resist- 

 ance of AB 2 , then must B 2 have n times the resistance of 

 B 2 to keep the potentials oi'Bx and B 2 equal. If condensers 

 be connected to the lines, as before mentioned, we require, 

 first, the resistance-balance of the last sentence applied to 

 every section between a pair of condensers ; and next, that 

 the capacity of a condenser in the line ABiO shall be, not 

 n times (as patented by Mr. Muirhead, I believe), but Ijn of 

 the capacity of the corresponding condenser in the line AB 2 G*. 

 * " On Duplex Telegraphy," Phil. Mag. January 1876. 



