348 On the Measurement of Alternate Currents, 



Another formula for cos e is 



._<v-c 



(8) 



cose= 4AB- < 5 > 



In the final formula (4) the factors of efficiency of the 

 separate coils (A, k) do not enter. This result depends, how- 

 ever, upon the fulfilment of the condition of parallelism 

 between the two coils. If the magnetic forces due to the 

 coils be inclined at different angles %, %' to the length of the 

 needle, we have in place of (3), 



C 2 = J (a cos x + v cos %') (a sin ^ + v sin %')<?£ 



=j[ia 2 sin2 % + ^ 2 sin2 % ' + ^sin( % + % 0]^; • (6) 

 while 



A 2 = isin 2 X j a 2 dt, B 2 = J sin tyjfrdt. . . (7) 

 Accordingly 



§av dt Q2 _ A2 - B 2 V {sin 2% . sin 2%'} 



{jV^xjVd*} ~ 2AB sin(x + %0 



in which the second fraction on the right represents the influ- 

 ence of the defect in parallelism. If % and %' are both nearly 

 equal to 45°, then approximately 



l/{sin2x.sin2 x '} ... (9) 



We have now to consider under what conditions the shunt- 

 current may be assumed to be proportional to the instantane- 

 ous value of the potential-difference at its terminals. The 

 obstacles are principally the self-induction of the shunt-coil 

 itself, and the mutual induction between it and the coil which 

 conveys the main current. As to the former, we know* that 

 if the mean radius of a coil be a, and if the section be circular 

 of radius c, and if n be the number of convolutions, 



L=Wa{k>g 8 ^-J} (10) 



To take an example from the shunt-coil used in the experi- 

 ments above referred to, where 



a=6cm., c=l cm., « = 32, 



L is of the order 10 5 cm. The time-constant of the shunt- 

 circuit (t) is equal to L/R, where R is the resistance in C.G.S. 



* Maxwell's < Electricity,' § 706, 



