Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/282

obtained by supposing that two electrons of charges e, e′, and velocities v, v′, possess electrokinetic energy of amount

${\frac {ee^{\prime }(\mathbf {v.v^{\prime }} )}{r}}+kee^{\prime }{\frac {d^{2}r}{ds\ ds^{\prime }}}vv^{\prime }$

Subtracting from this the mutual electrostatic potential energy, which is ee′c²/r, we may write the mutual kinetic potential of the two electrons in the form

${\frac {ee^{\prime }}{r}}({\dot {x}}{\dot {x}}^{\prime }+{\dot {y}}{\dot {y}}^{\prime }+{\dot {z}}{\dot {z}}^{\prime }-c^{2})+kee^{\prime }{\frac {d^{2}r}{ds\ ds^{\prime }}}vv^{\prime }$

where (x, y, z) denote the coordinates of e, and (x′, y′, z′) those of e′.

The unknown constant k has clearly no influence so long as closed circuits only are considered: if k be replaced by zero, the expression for the kinetic potential becomes

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

which, as will appear later, closely resembles the corresponding expression in the modern theory of electrons.

Clausius' formula has the great advantage over Weber's, that it does not compel us to assume equal and opposite velocities for the vitreous and resinous charges in an electric current; on the other hand, Clausius' expression involves the absolute velocities of the electrons, while Weber's depends only on their relative motion; and therefore Clausius' theory requires the assumption of a fixed aether in space, to which the velocities v and v′ may be referred.

When the behaviour of finite electrical systems is predicted from the formulae of Weber, Riemann, and Clausius, the three laws do not always lead to concordant results. For instance, if a circular current be rotated with constant angular velocity round its axis, according to Weber's law there would be a development of free electricity on a stationary conductor in the neighbourhood; whereas, according to Clausius' formula there would be no induction on a stationary body, but electrification