/ ciM.RAi.iy.AiioN or run Ri:Cirh-(>i.u. tiirokfm .w 



Now in \ie\v of the fori-^oin^; assuni|iiions and ri-slrictions wliitli 

 underlie all the pr(M)fs of the Reciprocal Theorem, known to the. 

 writer, it is by no means obvious that the theorem is valid when we 

 have to do with currents in continuous media as well as in linear 

 circuits, and when, furthermore we have to lake account of radiation 

 phenomena.' The proof or disproof of the theorem in the electro- 

 magnetic case is, howe\er, extremely important. The writer there- 

 fore, ofTers the following generalized Reciprocal Theorem, subject to 

 the restriction noted below. 



II. Let a distribution of impressed periodic electric intensity 

 F' =F\x. y, z) produce a corresponding distribution of current in- 

 tensity u' = u'{x, y, z), and let a second distribution of eqni-periodic 

 impressed electric intensity F" =F"{x, y, z) produce a second distribution 

 of current intensity ii" = u"(.v, y, s), tken 



((F'-u"hW = j\F"-u')dv, (6) 



the volume integration being extended o\er all conducting and 

 dielectric media. F and U are vectors and the expression [F-U) 

 denotes the scalar product of the two vectors. 



The only serious restriction on the generality of this theorem, as 

 proved below, is that magnetic matter is excluded: in other words it 

 is assumetl that all conducting and dielectric media in the field have 

 unit permeability. This restriction is theoretically to be regretted, 

 but is not of serious consequence in important practical applications. 



F'ROOF of tiENER-^LIZUD ReCIPRUCAL THEOREM^ 



In order to prove the generalized theorem stated above it is neces- 

 sary to discard the special assumption of quasi-stationary systems 

 underlying Rayleigh's theorem, and start with the fundamental 

 eciuations of electromagnetic theory. These ma>' be formulated as 

 follows: 



div B = 0. 



div E = 4irp. 

 curl E=-i|B, 



curl B = 4Tru + -^E. 

 where c is the velocity of light. 



' The theory of quasi-stationary systems expressly exiludes radiation. 

 * In the following proof it is necessary to assume a knowledge on the part of the 

 reader of the elements of vector analysis; the notation is that employed by .'\hraham. 



