Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/441

kinetic potential which concerns any one of them—say, e—may be written

$L_{e}=e(a_{x}{\dot {x}}+a_{y}{\dot {y}}+a_{z}{\dot {z}}-c^{2}\phi )$,

where a and φ denote potential functions, defined by the

$\mathbf {a} =\iiint {\frac {\rho ^{\prime }\mathbf {v} ^{\prime }}{r}}\ dx^{\prime }\ dy^{\prime }\ dz^{\prime },\qquad \phi =\iiint {\frac {\rho ^{\prime }}{r}}\ dx^{\prime }\ dy^{\prime }\ dz^{\prime }$;

ρ denoting the volume-density of electric charge, and v its velocity, and the integration being taken over all space.

We shall now reject Clausius' assumption that electrons act instantaneously at a distance, and replace it by the assumption that they act on each other only through the mediation of an aether which fills all space, and satisfies Maxwell's equations, This modification may be effected in Clausius' theory without difficulty; for, as we have seen,^[1] if the state of Maxwell's aether at any point is defined by the electric vector d and magnetic vector h,^[2] these vectors may be expressed in terms of potentials a and φ by the equations

$\mathbf {d} =c^{2}{\text{grad }}\phi -\mathbf {\dot {a}} ,\qquad \mathbf {h} ={\text{curl }}\mathbf {a}$;

and the functions a and φ may in turn be expressed in terms of the electric charges by the equations

$\mathbf {a} =\textstyle \iiint \{({\bar {\rho \mathbf {v} _{x}}})^{\prime }/r\}\ dx^{\prime }\ dy^{\prime }\ dz^{\prime },\qquad \phi =\iiint \{({\bar {\rho }})^{\prime }/r\}\ dx^{\prime }\ dy^{\prime }\ dz^{\prime }$,

where the bars indicate that the values of (ρv_x)′ and (ρ)′ refer to the instant (t - r/c). Comparing these formulae with those given above for Clausius' potentials, we see that the only change which it is necessary to make in Clausius' theory is that of retarding the potentials in the way indicated by L. Lorenz.^[3] The electric and magnetic forces, thus defined in terms of the

1. Cf. pp. 298, 299.
2. We shall use the small letters d and h in place of E and H, when we are concerned with Lorentz' fundamental case, in which the system consists solely of free aether and isolated electrons.
3. Cf. p. 298.