1891.] Application to Periodic Electric Currents. 209 



In this equation e t , r t are essentially positive, while e 2 , r 2 may be 

 either positive or negative. If e t and e 2 are both at disposal, the 

 minimum of (10), corresponding to the maximum current, is found 

 by making 



ei = 0, 62 = r-i ............ (11). 



But this is not the practical question. As in the case of simple 

 resistances, what we have to aim at is not to render the current in the 

 bridge a maximum, but rather the effect of the current. Whether 

 the receiving instrument be a galvanometer or a telephone, we cannot 

 in practice reduce its resistance to zero without at the same time 

 nullifying the effect desired. We must rather regard the space 

 available for the windings as given, and merely inquire how it may 

 best be utilised. Now the effect required to be exalted is, cceteris 

 paribus, proportional to the number of windings (m) ; and, if the 

 space occupied by insulation be proportional to that occupied by 

 copper, the resistance varies as m 2 . So also does the inductance ; and 

 accordingly, if the instrument be connected to the bridge by leads 

 sensibly devoid of resistance and inductance, 



(12), 



where ej, e 2 are independent of m. The quantity whose modulus is to 

 be made a minimum by variation of m is thus 



i (r 2 + m z e 2 ) . 



m m 



and we have 







This is a minimum by variation of m when 



Mod (! + -,) = Mod (eJi+tez) .......... (13). 



We may express this result by saying that to get the best effect 

 the instrument must be so wound that its impedance is equal to that 

 of the compound conductor r^ + ir z . If for any reason the inductances 

 can be omitted from consideration, then the resistance of the instru- 

 ment is to be made equal to rj, in accordance with Schwendler's 

 rale. 



The case of the "battery" branch may often be treated in like 

 manner. As Mr. Heaviside has shown, if a number of cells are 



