A Generalization of the Reciprocal Theorem

HE Reciprocal Theorem, an interesting a n d extremely imp o r t a n t relation of wide applicability, which was discovered by Lord Rayleigh, is stated by him in the language of electric circuit theory as follows: " L e t there be two circuits of insulated wire A and B, and in their neighborhood a n y combination 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 . " 1 Before proceeding with the generalization which is the subject of this paper, Rayleigh's theorem, in the following modified form, will first be stated and proved: I. Let a set of electromotive forces V . . . . Vn', all of the same frequency, acting in the n branches of an invariable network, produce a current distribution 1/ . . . . I,,', and let a second set of electromotive forces Vi" . . . . V,," of the same frequency produce a second current distribution I" . . . . In". Then (1) j-1 J=1 To prove this theorem we s t a r t with the equations of the network 2 ZjkIk=Vh k= .7 = 1 , 2 , . . . « , (2)