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 V. The advantages arising from the formulation of the world-postulate are illustrated by nothing so strikingly as by giving the expressions for the reactions exerted by a point-charge moving in any manner according to the theory. Let us conceive the world-line of such a point-like electron with the charge e, and let us introduce upon it the proper-time $$\tau$$ reckoned from any initial point. In order to obtain the field caused by the electron at any world-point $$P_{1}$$, let us construct the fore-cone belonging to $$P_{1}$$ (vide fig. 4). Evidently this cuts the unlimited world-line of the electron at a single point P, because these directions are all time-like vectors. At P, let us draw the tangent to the world-line, and let us draw from $$P_{1}$$ the normal $$P_{1}Q$$ to this tangent. Let r be the sum of $$P_{1}Q$$. According to the definition of a fore-cone, $$r/c$$ is to be reckoned as the sum of PQ. Now at the world-point $$P_{1}$$, the vector with respect to PQ of magnitude $$e/r$$ in its components along the x-, y-, z-axes, is represented by the vector-potential of the field multiplied by c; the component along the t-axis is represented by the scalar-potential of the field excited by e. This is the elementary law found out by, and.

In the description of the field caused by the electron itself, then it will appear that the division of the field into electric and magnetic forces is a relative one with respect to the time-axis assumed; the two forces considered together can most vividly be described by a certain analogy to the force-screw in mechanics; the analogy is, however, imperfect.

I shall now describe the ponderomotive force which is exerted by a point-charge moving in an arbitrary way, to another point-charge moving in an arbitrary way. Let us suppose