A Generalization of the Reciprocal Theorem 



By JOHN R. CARSON 



THK Rivi(>r<>i-al Tln'ori'm, an iiiltTcstinj; and extremely im- 

 portant relation of wide applical)ilit\'. which was disro\cred l)y 

 Lord Rayieiijh, is stated 1)\- him in the lanyjiiajje of electric circuit 

 theory as follows: 



"Let there lie two circuits of insulated wire A and B, and in their 

 neij;hlM)rhoo<l any comWinalion of wire circuits or solid conductors 

 in communication with condensers. A periodic electromotive force 

 in the circuit A will give rise to the same current in B as would be 

 excited in A if the electromotive force operated in B." ' 



Before proceeding with the generalization which is the subject of 

 this paper, Rayleigh's theorem, in the following modilied form, will 

 first be stated and proved : 



I. Let a set of electromotive forces Vi' .... V„', all of the same fre- 

 quency, acting in the n branches of an invariable network, produce a 

 current distribution Ii . . . . I„', and let a second set of electromotive 

 forces Vi" .... V\" of the same frequency produce a second current 

 distribution I \" .... /„". Then 



%Vjij'=%vri;. (1) 



To |)ro\e this theorem we start with the efjuations of the lu-twork 



^Z,kIk=V,. j=\.2....n. (2) 



and obser\e that, provide<i the network is invariable, contains no 

 internal source of energy or unilateral device, and provided that the 

 applied electromotive forces V\ . . . Vn are all of the same frequency, 

 say w'It, the mutual impedances satisfy the reciprocal relations 

 Zjk = Zk,- Consequently if (2) is sol\-ed for the currents, we get 



I,= %\jkVk. j=l,2...;;, (3) 



and the coefficients aI.so obey the reciprocal relations Ajk = Akj- 



Now consider two independent and arbitrary sets of equi-periodic 



applied electromotive forces, I'l' .... V„' and l'," .... !'„": then 

 ' Raylfigh, Thcorv- of Sound, Vol. I, p. 1.S5. 

 393 



