20 Equations of Propagation of Electric Waves. 



radiation : — The sources of one type are similar to infinitesimal Hertzian* 

 vibrators, being related in the same way to an axis, but the dependence' 

 of the emitted radiation on time is arbitrary ; the sources of the other 

 type are obtained therefrom by interchanging the roles of the electric 

 and magnetic forces. 



Another way of integrating the equations is to seek to express the- 

 values of the vectors, at one place and time, in terms of their values,, 

 at other places and times. The model for all investigations of thi& 

 kind is Green's Theory of the Potential. The main steps are (1) the- 

 determination of particular solutions, which tend to become infinite, in 

 definite ways, in the neighbourhood of chosen points ; (2) the discovery 

 of a theorem of reciprocity, connecting the values, on any chosen 

 surfaces, of two sets of solutions ; (3) the determination of the limiting, 

 form, assumed by the theorem of reciprocity, when the solutions of one 

 system have the assigned character of infinity at a given point. The* 

 result is the expression of the values of the functions of the other 

 system, at that point, and at a chosen instant of time, in terms of their 

 values, at all points on an arbitrary surface, and at determinate instants 

 of time. In the present theory, the solutions required for the first step 

 are among those alreadj 7 found ; the theorem of reciprocity is obtained 

 by a modification of the process by which the fundamental equations 

 can be deduced from the Action principle ; and the limiting form of the 

 theorem is found by adapting a process due to Kirchhoff. The result 

 is that the radiation which arrives at a chosen point may be regarded 

 as due to a distribution of imagined sources of radiation upon an 

 arbitrary closed surface, separating the point from all the aetual sources- 

 of radiation. The imagined sources are of the two types previously 

 specified ; and the directions of their axes, and the intensities of the 

 radiation sent out from them, are determined simply and directly by 

 the values, on the surface, of the vectors involved in the propagation 

 of the waves. A method for replacing the imagined sources of either 

 type by sources of the other type is indicated. The general theorem 

 is verified by choosing, for the arbitrary surface and the point, a sphere 

 and its centre; it then becomes equivalent to Poisson's well-known 

 solution of the differential equation of wave propagation in terms of 

 initial values. The " law of disturbance in secondary waves," to which 

 the theorem would give rise, is also determined ; it is, in essentials, 

 the same as has been found by previous writers. 



The general theorem is applied to the problem of the passage of 

 radiation through an aperture. When a train of radiation comes to a 

 perforated screen, or when electric vibrations take place in the dielectric 

 on one side (the nearer side) of a conducting surface, in which there is 

 an aperture, waves are sent out into the medium on the farther side ; 

 but the aperture also has the effect of generating a system of standing 

 waves on the nearer side. These systems of waves become, to a great 



