378 Lord Rayleigh on Forced Harmonic 



There is an interesting distinction to be noted here de- 

 pendent upon the manner in which the connections are made. 

 Consider, for example, the case of a bundle of five contiguous 

 wires wound into a coil, of which three wires connected in 

 series (so as to give maximum self-induction) constitute one 

 of the branches in parallel, and the other two, connected 

 similarly in series, constitute the other branch. There is still 

 an alternative in respect to the manner of connection of the 

 two branches. If steady currents w r ould circulate opposite 

 ways (M negative), the total current is divided into two parts 

 in the ratio of 3 : 2, in such a manner that the more powerful 

 current in the double wire nearly neutralizes at external 

 points the magnetic effects of the less powerful current in 

 the triple wire, and the total energy of the system is very 

 small. But now suppose that the connections are such that 

 steady currents would pass the same way round in both 

 branches (M positive). It is evident that the condition of 

 minimum energy cannot be satisfied if the currents are in 

 the same direction, but requires that the smaller current in 

 the triple wire should be in the opposite direction to the 

 larger current in the double wire. In fact the ratio of cur- 

 rents must be 3 : — 2; so that (as on the same scale the total 

 current is 1) the component currents in the branches are 

 both numerically greater than the total current which is 

 divided between them. And this peculiar feature becomes 

 more and more strongly marked the nearer L and N ap- 

 proach to equality*. 



When there are several conductors in parallel, the results 

 would in general be very complicated. When, however, 

 there is no mutual induction between the various members, a 

 simplification occurs. If the currents be denoted by yjr l7 ijr 2 , 

 ^ 3 . . . , the difference of potentials at the common ter- 

 minals is 



E = (vpL 1 -\-U 1 )'sjr 1 = (ipL 2 + R 2 )^r 2 = . . . . y 

 so that 



E 1 



f ++*+... SGpL + B)-* 



But if R/ and L' be the effective resistance and self-induction 

 respectively of the combination, 



" ; = R' + 2>I/ ; 



^1 + ^2 + ••■ 



* The reader who is interested in this subject is referred to my papers 

 in the Phil. Mag. July 1869, June 1870, "On Some Electromagnetic 

 Phenomena," &c. 



