I. The integration of the equations of propagation of electric waves

Abstract

1. In the older forms of the Undulatory Theory of Light, the propagation of the waves was traced by means of Huygens’ principle; each element of a wave front was regarded as becoming a source of disturbance from which secondary waves are emitted. The principle is indefinite, inasmuch as the nature and intensity of the sources of secondary waves are unrestricted, save by the conditions that the secondary waves must combine in advance so as to give rise to the disturbance actually pro­pagated, and must interfere in rear so as to give rise to no disturbance. That these conditions are insufficient, for the complete determination of the nature and intensity of the sources in question, is proved by observing that different writers, proceeding by different methods, have arrived at different expressions for “ the law of disturbance in secondary waves,” all these expressions satisfying the imposed conditions. In the more modern forms of the theory, the propagation of the waves is traced by means of a system of partial differential equations. This system has the same form, whether we regard the luminiferous medium as similar in its mode of action to an elastic solid, transmitting transverse waves, or regard light as an electromagnetic disturbance obeying the fundamental equations of the electric field. In both cases it appears that all the components of the vector quantities which represent the dis­turbance satisfy a partial differential equation of the form ∂ 2 ϕ/∂ l 2 = C 2 ∇ 2 ϕ.