## Extract

By point-potential in this paper is meant the potential scalar or vector of a point-charge which is moving in any manner without any restriction except that the speed is less than that of light. Its general form is *f*(τ)/*r*(1 — C^{-1}∂*r*/∂τ), where *r* is the distance of the point-charge at a time τ from the field-point, C the speed of light, and *f*(τ) an arbitrary function of τ, τ being given by the equation C(*t* — τ) = *r*. The solution for uniform rectilinear motion of the point-charge was given by Sir J. J. Thomson and Heaviside. The solution for a general motion of the point-charge was given by Liénard and Wiechert. A method of deducing the result by complex integration was given by the writer, and this method is the basis of the present paper. The subsequent literature is extensive and falls, for the most part, under two heads. First, calculations of electromagnetic inertia include, for uniform motion, many papers by Heaviside, Searle, Morton, Abraham and others. For variable motion a typical paper is Sommerfeld. A Lagrangian expansion appears to have been given first by Herglotz, by Schott, and in two papers by the author. The second head contains calculations of the radiation and of the forces at a considerable distance from the origin. Many results were given by Heaviside, principally in letters to ‘ Nature.' His work appears to have escaped notice by most writers on the subject. Radiation from circular and elliptic orbits was given by him in 'Nature,' October 30 and November 16,1908. Similar subjects were treated by Schott. Reference should be made for fuller treatment to Vol. Ill of Heaviside’s ' Electromagnetic Papers,' and to Schott’s ‘Electromagnetic Radiation.’ The latter work contains many references to this part of the subject. In this paper a new type of expansion is studied. Let O be any origin of axes having fixed directions in space, and let E be tbe position of the point-charge, P being the field-point. Then the point-potential is expanded in a series ΣY_{n}U_{n}, where Y_{n} is a spherical harmonic of the angles made by OP with the axes and U_{n} is a function depending on the positions of O and E and the distance OP. It is thus analogous to the well-known harmonic expansions of *r*_{-1} or of the more general potential *e*^{-kr}*r*^{-1}. It is, however, of a more general type, for the origin is unrestricted as to its motion ; it may be at rest or it may move with the point-charge or in any other manner whatever. As, in addition, the point-charge is unrestricted in motion except that its speed is less than that of light, it is clear that the investigation of the functions U_{n} is a very wide one. In this paper the convergence of the series is investigated and methods given for calculating them. As an example a formal solution of the problem of the motion of a point-charge outside a fixed spherical conductor is given. In the last section the connection with the Lagrangian expansion is given.

## Footnotes

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- Received February 11, 1918.

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