Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/376

It is evident that the quantity a thus defined satisfies the equation

$\nabla ^{2}\mathbf {a} =\nabla ^{2}\mathbf {a} _{1}+{\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}\mathbf {a}$,

or $\nabla ^{2}\mathbf {a} -{\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}\mathbf {a} =-4\pi {\boldsymbol {\iota }}$. This equation may be written

${\text{curl }}\mathbf {H} =(1/c^{2})\mathbf {\dot {E}} +4\pi {\boldsymbol {\iota }}$,

while the equations H = curl a, $\mathbf {E} =-\mathbf {\dot {a}}$ give

${\text{curl }}\mathbf {E} =-\mathbf {\dot {H}}$.

These are, however, the fundamental equations of Maxwell's theory in the form given in his memoir of 1868.^[1]

That Hertz's deduction is ingenious and interesting will readily be admitted. That it is conclusive may scarcely be claimed: for the argument of Helmholtz regarding the induction of currents is not altogether satisfactory; and Hertz, in following his master, is on no surer ground.

In the course of a discussion^[2] on the validity of Hertz's assumptions, which followed the publication of his paper, E. Aulinger^[3] brought to light a contradiction between the principles of the unity of electric and of magnetic force and the electrodynamics of Weber. Consider an electrostatically charged hollow sphere, in the interior of which is a wire carrying a variable current. According to Weber's theory, the sphere would exert a turning couple on the wire; but according to Hertz's principles, no action would be exerted, since charging the sphere makes no difference to either the electric or the magnetic force in its interior. The experiment thus suggested would be a crucial test of the correctness of Weber's theory; it has the advantage of requiring nothing but closed currents and electrostatic charges at rest; but the quantities to be observed would be on the limits of observational accuracy.

1. Cf. p. 287.
2. Lorberg, Ann. d. Phys. xxvii (1886), p. 666; xxxi (1887), p. 131. Boltzmann, ibid. xxix (1886), p. 598.
3. Ann. d. Phys. xxvii (1886), p. 119.