Page:Aether and Matter, 1900.djvu/209



. results above obtained have been derived from the correlation developed in § 106, up to the first order of the small quantity v/c, between the equations for aethereal vectors here represented by (f', g', h') and (a', b', c') referred to the axes (x', y', z') at rest in the aether and a time t", and those for related aethereal vectors represented by (f, g, h) and (a, b, c) referred to axes (x', y', z') in uniform translatory motion and a time t'. But we can proceed further, and by aid of a more complete transformation institute a correspondence which will be correct to the second order. Writing as before t" for $$t'-\frac{v}{c^{2}}\epsilon x'$$, the exact equations for (f, g, h) and (a, b, c) referred to the moving axes (x', y', z') and time t' are, as above shown, equivalent to

$$\begin{array}{ccc} 4\pi\frac{df'}{dt}=\frac{dc'}{dy'}-\frac{db'}{dz'} & & -\left(4\pi c^{2}\right)^{-1}\frac{da'}{dt}=\frac{dh'}{dy'}-\frac{dg'}{dz'}\\ \\4\pi\epsilon\frac{dg'}{dt}=\frac{da'}{dz'}-\frac{dc'}{dx'} & & -\left(4\pi c^{2}\right)^{-1}\epsilon\frac{db'}{dt}=\frac{df'}{dz'}-\frac{dh'}{dx'}\\ \\4\pi\epsilon\frac{dh'}{dt}=\frac{db'}{dz'}-\frac{da'}{dy'} & & -\left(4\pi c^{2}\right)^{-1}\epsilon\frac{dc'}{dt}=\frac{dg'}{dx'}-\frac{df'}{dy'}.\end{array}$$

Now write

$$\left(x_{1},\ y_{1},\ z_{1}\right)$$ for $$\left(\epsilon^{\frac{1}{2}}x',\ y',\ z'\right)$$

$$\left(a_{1},\ b_{1},\ c_{1}\right)$$ for $$\left(\epsilon^{-\frac{1}{2}}a',\ b',\ c'\right)$$ or $$\left(\epsilon^{-\frac{1}{2}}a,\ b+4\pi vh,\ c-4\pi vg\right)$$