168 Drs. Crehore and Squier on the Currents 



inductance L 3 be the independent variable, and let it be 

 required to calculate the value of L 4 for zero current in the 

 galvanometer. 



Denote the reactance L 3 &> by x, and the reactance L 4 « by y, 

 then equation (11) may be written in the form 



i= *±£ = £+2?, ( 



n -\-x 2 r + sx 7 v 



where k is a constant equal to y£- = -P2.T2 2 > ancl m = ^ ; 



-n = Rf; j9=K 3 K 4 ; ^=L 2 ft) ; r^R^; 5=^0). 



Equation (12) may be reduced to 



qx 2 y — sxy 2 +px 2 — ry 2 + ex +fy + # = 0, . (13) 

 where 



e=— ms=— R^I^G) ; / = ^m=R|L 2 ©; 



and 



# =pn-mr=B. 2 R 4 Rl-n i R 3 Rl. 



This is evidently an equation of the third degree between 

 two variables, the reactances of the coils, and it enables one 

 to find the inductance y of one coil which will just balance a 

 given inductance x in the other. 



This equation is of use in calculating points of maximum 

 sensitiveness, and it also shows other useful relations. For 

 instance, if the resistances 1^ andR 2 are made approximately 

 equal as well as the inductances L x and L 2 , the equation 

 shows that not only is it necessary for the inductances of the 

 other two circuits (3) and (4) to be equal for zero current, but 

 the resistances of those branches must also be equal. Sub- 

 stituting x=y in (13) satisfies the equation if R 1 = R 2 ; 

 R 3 = R 4 ; and L 3 =L 4 . 



This point may be clearly seen in the diagrams. In fig. 3, 

 in which the points P and Q of fig. 2 are merely brought into 

 coincidence, it will be seen that 



tanPOH=^ ? (14) 



tanPOK=^, (15) 



tanAPM=fe (16) 



■"3 



tanAPN=^ . (17) 



