488 Dynamical Theory of Currents [CH. xvi 



There are no batteries and no condenser in the arms in which the 

 currents ^ and i, flow. The currents are, however, connected by the 



relation 



ii + is^I .............................. (520), 



so that the corresponding coordinates x-^ and x. 2 are connected by 



The dynamical equations are now found to be (cf. equation (519)), 

 ~(Z^ 1 



If we subtract and replace i. 2 by / - ^ from equation (520), we eliminate 

 X and obtain 



(L + N-2M)^ t +(M-N)^ = SI-(R-S)i L ...... (521). 



If / is given as a function of the time, this equation enables us to deter- 



mine i lt and thence i 2 . 



ose tha 

 ut 2=i 



S-(M-N)ip 



t . 



For instance, suppose that the current / is an alternating current of 

 frequency p. If we put 2=i e ipt , the solution of equation (521) is 





R (M L)ip 

 while similarly 



(L + N ^ m)ip + ( R +S) 



When p = 0, the solution of course reduces to that for steady currents. 

 As p increases, we notice that the three currents i lt i 2 and I become, in 

 general, in different phases, and that their amplitudes assume values 

 which depend on the coefficients of induction as well as on the resistances. 

 Finally, for very great values of p, the values of ^ and ^' 2 are given by 



N -M~~ L-M~ L+N-2M ' 



shewing that the currents are now in the same phase and are divided in a 

 ratio which depends only on their coefficients of induction. For instance, 

 if the arms ACB, ADE are arranged so as to have very little mutual 

 induction (M very small), the current will distribute itself between the 

 two arms in the inverse ratio of the coefficients of self-induction. 



It is possible to arrange for values for Z, M and N such that the two 

 currents i\ and i 2 shall be of opposite sign. In such a case the current in one 

 at least of the branches is greater than that in the main circuit. Let us, for 

 instance, suppose that the branches consist of two coils having r and s turns 



