100 BELL SYSTEM TECHNICAL JOURNAL 



same field as that produced by given sources inside C; and also the 

 field inside C is the same as that produced by given sources outside C. 

 One of these systems of sources can be identically equal to zero. 



The actual calculations are made as follows. From the discon- 

 tinuities in the tangential components of E and H, we obtain / and 

 Af by (13) and (14). From these currents we find the two vector 

 potentials 



'Ax',y' 



47rJ J 



(C) 



4xJ J 



Mf^^3^),-..,5, 



e-»^^ dS, 



(16) 



(C) 



where r = yj{x — x'y -{- {y — y'Y + (2 — 2')^ is the distance between 

 a point P{x, y, z) somewhere in space and a point P'(x', y', 2') on C. 

 From these potentials we calculate the electric intensity and the 

 magnetic intensity by 



„ . . , grad div A . „ 



E = — toifjL A + -. curl F, 



tcae 



, . grad div 7^ . 

 H = curl A + ^ ■ t^e F. 



(17) 



The proof of the Equivalence Principle can be modified so as to 

 throw some additional light on it. Let us suppose that given sources 

 S' are inside the closed surface C and let us make our new synthetic 

 field by obliterating the old field outside C and leaving everything as 

 it was inside C. The new field has the same sources S' and besides it 

 is discontinuous across C. These discontinuities are the additional 

 sources 5 whose densities are calculable from (13) and (14). Since 

 the new field is identically zero outside C, the field produced by 5 is 

 such as to cancel the field produced by S' outside C. Thus the system 

 of sources 5 acts as a perfect absorber for the electromagnetic wave 

 produced by S' . Reversing the directions of the current distributions 

 on C, we conclude that the system of sources — 5 produces outside 

 C exactly the same field as S' . 



The Equivalence Principle is closely related to another theorem 

 which we may call the Induction Theorem. Let us suppose that a 

 closed surface C subdivides the entire space into two homogeneous 

 media and that a system of sources 5 is given in one of those regions 

 (Fig. 4). Let E, II be the field due to these sources on the assumption 

 that the medium inside C is the same as that outside. The true field 



