Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/251

the conductor is one to which, for various reasons which will appear later, objection may be taken; but it is an integral part of Weber's theory, and cannot be excised from it. In fact, if this condition were not satisfied, and if the law of force were Weber's, electric currents would exert forces on electrostatic charges at rest^[1]; as may be seen by the following example. Let a current flow in a closed circuit formed by arcs of two concentric circles and the portions of the radii connecting their extremities; then, if Weber's law were true, and if only one kind of electricity were in motion, the current would evidently exert an electrostatic force on a charge placed at the centre of the circles. It has been shown,^[2] indeed, that the assumption of opposite electricities moving with equal and opposite velocities in a circuit is almost inevitable in any theory of the type of Weber's, so long as the mutual action of two charges is assumed to depend only on their relative (as opposed to their absolute) motion.

The law of Weber is not the only one of its kind; an alternative to it was suggested by Bernhard Riemann (b.1826, d. 1866), in a course of lectures which were delivered^[3] at Göttingen in 1861, and which were published after his death by K. Hattendorff, Riemann proposed as the electrokinetic energy of two electrons e(x, y, z) and e′(x′, y′, z′) the expression

$-{\frac {1}{2}}{\frac {ee^{\prime }}{r}}\{({\dot {x}}-{\dot {x}}^{\prime })^{2}+({\dot {y}}-{\dot {y}}^{\prime })^{2}+({\dot {z}}-{\dot {z}}^{\prime })^{2}\}$;

this differs from the corresponding expression given by Weber only in that the relative velocity of the two electrons is substituted in place of the component of this velocity along the radius vector. Eventually, as will be seen later, the laws

1. This remark was first made by Clausius, Journal für Math. lxxxii (1877), p. 86: the simple proof given above is due to Grassmann, Journal für Math. lxxxiii (1877), p. 57.
2. H. Lorberg, Journal für Math. lxxxiv (1878), p. 300.
3. Schwere, Elektricität und Magnetimus, nach den Vorlesungen von B. Riemann: Hannover, 1875, p. 326. Another alternative to Weber's law had been discovered by Gauss so far back as 1835, but was not published until after his death: cf. Galug' Works, v, p. 616.