Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/461

easily be derived by considering that the equation of the rectangular hyperbola

$x^{2}-(ct)^{2}=1$

(in the plane of the variables x, ct) is unaltered when any pair of conjugate diameters are taken as new axes, and a new unit of length is taken proportional to the length of either of these diameters. The equations of transformation are thus found to be

${\begin{matrix}x&=&x_{1}\cosh a&+&ct_{1}\sinh a,&\qquad &y&=&y_{1},\\t&=&t_{1}\cosh a&+&(x_{1}/c)\sinh a,&\qquad &z&=&z_{1},\end{matrix}}$

where a denotes a constant. The simpler equations previously given by Lorentz^[1] may evidently be derived from these by writing w/c for tanh a, and neglecting powers of w/c above the first. By an obvious extension of the equations given by Lorentz for the electric and magnetic forces, it is seen that the corresponding equations in the present transformation are

${\begin{cases}d_{x}&=&d_{x_{1}},\\d_{y}&=&d_{y_{1}}\cosh a+ch_{z_{1}}\sinh a,\\d_{z}&=&d_{z_{1}}\cosh a-ch_{y_{1}}\sinh a,\end{cases}}$

${\begin{cases}h_{x}&=&h_{x_{1}},\\h_{y}&=&h_{y_{1}}\cosh a-(1/c)d_{z_{1}}\sinh a,\\h_{z}&=&h_{z_{1}}\cosh a+(1/c)d_{y_{1}}\sinh a,\end{cases}}$

The connexion between ρ and ρ₁ may be obtained in the following way. It is assumed that if a charge e is attached to a particle which occupies the position (ξ, η, ζ) at the instant t, an equal charge will be attached to the corresponding point (ξ₁, η₁, ζ₁) at the corresponding instant t₁, in the transformed system; so that a charge e′ attached to an adjacent particle (ξ + Δξ, η + Δη, ζ + Δζ) at the instant t will give rise in the derived system to a charge e′ at the place

$\left(\xi _{1}+{\frac {\partial \xi _{1}}{\partial \xi }}\Delta \xi +{\frac {\partial \xi _{1}}{\partial \eta }}\Delta \eta +{\frac {\partial \xi _{1}}{\partial \zeta }}\Delta \zeta ,\qquad \eta _{1}+{\frac {\partial \eta _{1}}{\partial \xi }}\Delta \xi +{\frac {\partial \eta _{1}}{\partial \eta }}\Delta \eta +{\frac {\partial \eta _{1}}{\partial \zeta }}\Delta \zeta ,\right.$

$\left.\zeta _{1}+{\frac {\partial \zeta _{1}}{\partial \xi }}\Delta \xi +{\frac {\partial \zeta _{1}}{\partial \eta }}\Delta \eta +{\frac {\partial \zeta _{1}}{\partial \zeta }}\Delta \zeta \right)$

at the instant

$\left(t_{1}+{\frac {\partial t_{1}}{\partial \xi }}\Delta \xi +{\frac {\partial t_{1}}{\partial \eta }}\Delta \eta +{\frac {\partial t_{1}}{\partial \zeta }}\Delta \zeta \right)$;

1. Cf. p. 434.