Page:Aether and Matter, 1900.djvu/203

 place until it is proved to be in definite contradiction, not removable by suitable modification, with another portion of it.

. We now recall the equations of the free aether, with a view to changing from axes (x, y, z) at rest in the aether to axes (x', y', z') moving with translatory velocity v parallel to the axis of x; so as thereby to be in a position to examine how phenomena are altered when the observer and his apparatus are in uniform motion through the stationary aether. These equations are

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

When they are referred to the axes (x', y', z') in uniform motion, so that $$(x',\ y',\ z')=(x-vt,\ y,\ z),\ t'=t$$, then $$d/dx,\ d/dy,\ d/dz$$ become $$d/dx',\ d/dy',\ d/dz'$$, but d/dt becomes $$d/dt'-vd/dx'$$: thus

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

where

We can complete the elimination of (f, g, h) and (a, b, c) so