376 Lord Rayleigh on Forced Harmonic 



the series. Thus a 13 , a u , a 24 , . . . (or, as we should here call 

 them, M 13 , M 14 , M 24 , . . .) are supposed to vanish, as would 

 usually happen in experiment. For the sake of distinctness 

 we will limit ourselves to four circuits. 



In the fourth circuit the current is due ex hypothesi only to 

 induction from the third. Its reaction upon the third, for 

 the rate of alternation under contemplation, is given at once 

 by (11) and (12) ; and if we use the complete values applicable 

 to the third circuit under these conditions, we may thence- 

 forth ignore the fourth circuit. In like manner we can now 

 deduce the reaction upon the secondary, giving the effective 

 resistance and self-induction of that circuit under the influ- 

 ence of the third and fourth circuits; and then, by another 

 step of the same kind, we may arrive at the values applicable 

 to the primary circuit, under the influence of all the others. 

 The process is evidently general ; and we know by the theorem 

 that, however numerous the train of circuits, the influence of 

 the others upon the first must be to increase its effective 

 resistance and diminish its effective inertia, in greater and 

 greater degree as the rapidity of alternations increases. 



In the limit, when the rapidity of alternation increases 

 indefinitely, the distribution of currents is determined by the 

 induction-coefficients irrespective of resistance, and it is of 

 such a character that the currents are alternately opposite in 

 sign as we pass along the series*. 



As another example under the head of two degrees of 

 freedom, we will take the case of two electrical conductors in 

 parallel. It is not necessary to include the influence of the 

 leads outside the points of bifurcation. Provided there be no 

 mutual induction between these parts and the remainder, their 

 induction and resistance enter into the result by simple 

 addition. 



Under the operation of resistance only, the total current ^ 

 would divide itself between the conductors R and S in the 

 parts 



s*» and R±l 



E+S, and K + 8- 

 We may conveniently take the second coordinate ^r 2 so that 

 the currents in the two conductors are 



S T? 



ITS^ 1 -^ 2 ' and B+S* 1- * 2 ' 

 ^ still representing the total current. 



* See a paper, "On some Electromagnetic Phenomena considered in 

 connection with the Dynamical Theory," Phil. Mag. July 1869. 



