INDUCTANCE AND CAPACITY 409 



and Heaviside showed that the observations obtained by means 

 of the Hughes apparatus are, when correctly reduced, in entire 

 accord with the accepted theory of induction. 



The connections for this form of bridge are shown in Fig. 238. 

 They are much like those for the Wheatstone bridge, but in the 

 galvanometer circuit is included the secondary of the air-core 

 transformer of variable ratio, the primary of this transformer 

 being connected in the lead running from the source of current 

 to the bridge. The mutual inductance of the air-core transformer 

 will be represented by m. Assuming sinusoidal currents the 

 mesh equations are: 



(X + Y) (Z x + Z P + Z G ) - XZ G - I B Z P - jmuI B = 

 X(Z M + Z G + Z N ) - (X + Y) Z G - I B Z N + jmuI B = 0. 



In respect to the sign given to the term involving the mutual 

 inductance, in this and other methods of measurement, it may 

 be either positive or negative depending on the manner in which 

 the device is connected into the circuit. However, the particular 

 connection and the corresponding sign in the equations must be 

 used which will enable a balance to be obtained. 



Solving the above equations for F, the galvanometer current, 

 and substituting the values of the impedances, the arms M, N 

 and P being non-inductive, 



+ R N ) - (R N -jmu)(R x + RP + jLx<*) 



denominator 



If only the condition of balance is required, it is not necessary to 

 know the expression for the denominator. For balance, the 

 numerator must be zero or 



HP (R M + R N ) - R N (R X + RP) - mrfL 



(Rx + R P ) mu - LxR N <] = 0. 



Separating the quadrature components, the horizontal compo- 

 nent gives 



R P R M - R N Rx = mrfLx 

 and the vertical component gives 



m(R M + RN + Rx + RP) = RNL X . 

 In order to obtain a balance both these equations must be satisfied. 



