Continuation of Solutions and the Retarded Electromagnetic Fields in the Mathematical Formulation of Huygens' Principle

In this paper we examine to what extent the solutions of Maxwell's equations proposed as formulations of Huygens' principle for these equations are indeed such. The formulae derived in a previous paper, as well as those derived by Larmor, by Jones and by Baker and Copson, express the actual field produced by a given charge and current distribution at any point interior to a closed surface in terms of certain retarded surface integrals of the actual field plus a particular field obtained from retarded volume integrals of the given charge and current distribution. We study this particular field and find that it satisfies Maxwell's equations, but for charge and current distributions other than those when the normal component of the given current at the surface is non-vanishing. We show how, in spite of this, the actual field satisfies Maxwell's equations for the given charge and current. We use the identity derived in the previous paper to study the ‘continuation of solutions’ first investigated by Larmor. Amongst other things, we show that the surface integral does not, in general, represent the effect on one side of S produced by the source on the other side. The assertion that the surface integral over S represents the effect produced on one side of the surface by the sources on the other side (Huygens' principle) is valid only ifthe normal component of the current density is zero at S, a proviso unrecognized in many books and in the literature.