Energetics of the Induction Balance. 57 



in which each arm consists o£ a coil and condenser in parallel. 

 Slightly greater generality is obtained, however, and more 

 practical methods are embraced in the general solution, if the 

 condenser is shunted across part of the coil only. In that 

 •case, if R and L denote the resistance and inductance 

 respectively of the whole coil, and r and I the corresponding- 

 values for that part shunting the condenser, then the voltage 

 across the coil terminals is determined by 



di 

 ^ L ^+ Pa '-^-^ (!) 



where i is the total current and q is the condenser charge at 

 that particular moment. Elimination of q from (1) at once 

 gives Z, the resistance operator (generalized resistance), for 

 this case:* 



Z = R-r+{L-l)'D+{CD + (r + lD)- 1 \-\ . . (2) 



where is the capacity of the condenser, and D is the time 



d 

 operator-,. From the expansion of Z^ — Z 2 Z 3 , the con- 

 dition to be fulfilled for a first order balance is found to be 

 R 4 (L l -r 1 2 C 1 )-R3(L 2 - } - 2 =C 2 )=R 2 (L 3 -r 3 2 C 3 ;-E 1 (L 1 -n 2 4 ). 



• • • (3; 



All practical methods of measurement are so chosen that no 

 second order effects occur; and hence, in such methods, true- 

 balance is uniquely determined by satisfying condition (ii.), 

 that is, the appropriate modification of equation (3 ). But 

 Heaviside's method of functional operators, though rigorous 

 and elegant, is not, as a rule, appreciated by the ordinary 

 student; and it is therefore desirable to have an alternative 

 method of the same general applicability. The object of the 

 present note is to show that simple energy considerations 

 lead to an equation equivalent to (3), and that the result 

 summarizes in a simple and convenient way the relationships 

 existing between the circuit constants in the state of balance 

 in all bridge methods of comparing inductances and capacities. 



§2. If, in equation (1), v is initially zero and eventually 

 reaches a steady value V, then integration with respect to 

 the time yields 



J 



(v— Ri)dt = IA— rQ, ...... (4) 



where Q and 1 are the final steady values of q and i 



