Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/460

traction. After this Lorentz^[1] went further still, and obtained the transformation in a form which is exact to all orders of the small quantity w/c. In this form we shall now consider it.

The fundamental equations of the aether are

${\begin{matrix}{\text{div }}\mathbf {d} &=&4\pi c^{2}\rho ,\qquad \qquad &{\text{curl }}\mathbf {d} &=&-\mathbf {\dot {h}} ,\\{\text{div }}\mathbf {h} &=&0,\qquad \qquad &{\text{curl }}\mathbf {h} &=&(1/c^{2})\mathbf {\dot {d}} +4\pi \rho \mathbf {v} ,\end{matrix}}$

It is desired to find a transformation from the variables x, y, z, t, ρ, d, h, v, to new variables x₁, y₁, z₁, t₁, ρ₁, d₁, h₁, v₁, such that the equations in terms of these new variables may take the same form as the original equations, namely:

${\begin{matrix}{\text{div}}_{1}\ \mathbf {d} _{1}&=&4\pi c^{2}\rho _{1},\qquad \qquad &{\text{curl}}_{1}\ \mathbf {d} _{1}&=&-\partial \mathbf {h} _{1}/\partial t_{1},&\ \\{\text{div}}_{1}\ \mathbf {h} _{1}&=&0,\qquad \qquad &{\text{curl}}_{1}\ \mathbf {h} _{1}&=&(1/c^{2})\partial \mathbf {d} _{1}/\partial t_{1}&+\ 4\pi \rho _{1}\mathbf {v} _{1},\end{matrix}}$

Evidently one particular class of such transformations is that which corresponds to rotations of the axes of coordinates about the origin: these may be described as the linear homogeneous transformations of determinant unity which transform the expression (x² + y² + z²) into itself.

These particular transformations are, however, of little interest, since they do not change the variable t. But in place of them consider the more general class formed of all those linear homogeneous transformations of determinant unity in the variables x, y, z, ct, which transform the expression (x² + y² + z - c²t²)^{[errata 1]} into itself: we shall show that these transformations have the property of transforming the differential equations into themselves.

All transformations of this class may be obtained by the combination and repetition (with interchange of letters) of one of them, in which two of the variables—say, y and z—are unchanged. The equations of this typical transformation may

1. Proc. Amsterdam Acad. (English ed.), vi, p. 809. Lorentz' work was completed in respect to the formulae which connect ρ₁, v₁ with ρ, v, by Einstein, Ann. d. Phys., xvii (1905), p. 891. It should be added that the transformation in question had been applied to the equation of vibratory motions many years before by Voigt, Gött. Nach, 1887, p. 41.

errata

1. Correction: (x² + y² + z - c²t²) should be amended to (x² + y² + z² - c²t²)