THE RECIPROCAL ENERGY THEOREM 329 



The foregoing may be summed up in the following proposition. 



The relative transmission figure of merit oj any two antennas with 

 respect to transmission from a given transmitting point to a given re- 

 ceiving point is equal to the ratio of their resistances as measured from 

 their operating branches, multiplied by the square of the ratio of their 

 receiving sensitivities with respect to transmission from the receiving 

 point to the transmitting point. 



This theorem has a considerable field of practical utility. For 

 example it enables us to deduce the relative transmitting properties 

 and efficiency of any antenna system from its receiving efficiency. 

 It has already been so applied in one actual case of large importance. 



Note on the Reciprocal Theorem 



The proof of the Reciprocal Theorem, as given originally by Lord 

 Rayleigh, was applicable only to 'quasi-stationary' transducers, that 

 is transducers which obey the simple laws of electric circuit theory. 

 In the July 1924 issue of the Bell System Technical Journal the writer 

 stated and proved a generalized theorem subject, however, to the 

 restriction that the permeability ix of the medium shall be everywhere 

 unity. The theorem referred to is 



Let a distribution of impressed periodic electric intensity F' = F'{x, y, z) 

 produce a corresponding distribution of current intensity u' = u'{x, y, z), 

 and let a second distribution of equi-periodic impressed electric intensity 

 F" = F"{x, y, z) produce a second distribution of current intensity 

 u" = u"{x, y, z), then 



f{F'-u")dv = f(F"-u')dv, 



the volume integration being extended over all conducting and dielectric 

 media. F and u are vectors and the expression (F-u) denotes the scalar 

 product of the two vectors. 



Later Pleijel ^ stated the theorem for unrestricted values of n. In 

 discussing reciprocal theorems in the June 1929 issue of the Proc. 

 I. R. E. the writer expressed some doubt as to the validity of Pleijel's 

 proof (which is entirely different from my own). Subsequent study, 

 however, has convinced me that except for minor and easily remedied 

 errors, the proof is entirely sound. Later the writer discovered that 

 the restriction n = 1 can easily be removed from his own original 

 proof as will now be shown.^ 



'"Two Reciprocal Theorems in Electricity," Ingeniors V'etenskaps Akademien 

 Nr. 68, 1927. 



* Another and somewhat different extension of the proof has been derived by my 

 associate Dr. W. H. Wise. 



