Page:Lorentz Simplified1899.djvu/3

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In most applications ${\mathfrak {p}}$ would be the velocity of the earth in its yearly motion.

§ 4. Now, in order to simplify the equations, the following quantities may be taken as independent variables

$x'={\frac {V}{\sqrt {V^{2}-{\mathfrak {p}}_{x}^{2}}}}x,\ y'=y,\ z'=z,\ t'=t-{\frac {{\mathfrak {p}}_{x}}{V^{2}-{\mathfrak {p}}_{x}^{2}}}x$.

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The last of these is the time, reckoned from an instant that is not the same for all points of space, but depends on the place we wish to consider. We may call it the local time, to distinguish it from the universal time t.

If we put

${\frac {V}{\sqrt {V^{2}-{\mathfrak {p}}_{x}^{2}}}}=k$,

we shall have

${\frac {\partial }{\partial x}}=k{\frac {\partial }{\partial x'}}-k^{2}{\frac {{\mathfrak {p}}_{x}}{V^{2}}}{\frac {\partial }{\partial t'}},\ {\frac {\partial }{\partial y}}={\frac {\partial }{\partial y'}},\ {\frac {\partial }{\partial z}}={\frac {\partial }{\partial z'}},\ {\frac {\partial }{\partial t}}={\frac {\partial }{\partial t'}}$.

The expression

${\frac {\partial {\mathfrak {A}}_{x}}{\partial x'}}+{\frac {\partial {\mathfrak {A}}_{y}}{\partial y'}}+{\frac {\partial {\mathfrak {A}}_{z}}{\partial z'}}$

will be denoted by

$Div'\ {\mathfrak {A}}$.

We shall also introduce, as new dependent variables instead of the components of ${\mathfrak {d}}$ and ${\mathfrak {H}}$, those of two other vectors ${\mathfrak {F}}'$ and ${\mathfrak {H}}$, which we define as follows

${\begin{array}{lllll}{\mathfrak {F}}'_{x}=4\pi V^{2}{\mathfrak {d}}_{x},&&{\mathfrak {F}}'_{y}=4\pi kV^{2}{\mathfrak {d}}_{y}-k{\mathfrak {p}}_{x}{\mathfrak {H}}_{z},&&{\mathfrak {F}}'_{z}=4\pi kV^{2}{\mathfrak {d}}_{z}-k{\mathfrak {p}}_{x}{\mathfrak {H}}_{y},\\\\{\mathfrak {H}}'_{x}=k{\mathfrak {H}}_{x},&&{\mathfrak {H}}'_{y}=k^{2}{\mathfrak {H}}_{y}+4\pi k^{2}{\mathfrak {p}}_{x}{\mathfrak {d}}_{z},&&{\mathfrak {H}}'_{z}=k^{2}{\mathfrak {H}}_{z}-4\pi k^{2}{\mathfrak {p}}_{x}{\mathfrak {d}}_{y}.\end{array}}$,

In this way I find by transformation and mutual combination of the equations (I_b)—(V_b):