Page:Elektrische und Optische Erscheinungen (Lorentz) 050.jpg

It is now

${\frac {\partial }{\partial x}}=\left({\frac {\partial }{\partial x}}\right)'-{\frac {{\mathfrak {p}}_{x}}{V^{2}}}{\frac {\partial }{\partial t'}}{\text{, etc.}}$

(35)

and

${\frac {\partial }{\partial t}}={\frac {\partial }{\partial t'}}{,}$

so that we find for the determination of ${\mathfrak {H}}'$

$V^{2}\Delta '{\mathfrak {H'}}_{x}-{\frac {\partial ^{2}{\mathfrak {H'}}_{x}}{\partial t'^{2}}}=4\pi V^{2}\left[\left\{{\frac {\partial (\rho {\mathfrak {v}}_{y})}{\partial z}}\right\}'-\left\{{\frac {\partial (\rho {\mathfrak {v}}_{z})}{\partial y}}\right\}'\right]{\text{, etc.}}$

A solution of these equations is easy to give. Namely, imagine three functions $\psi _{x}$, $\psi _{y}$, $\psi _{z}$ that satisfy the conditions

$V^{2}\Delta '\psi _{x}-{\frac {\partial ^{2}\psi _{x}}{\partial t'^{2}}}=4\pi V^{2}\rho {\mathfrak {v}}_{x}{\text{, etc.}}$

(36)

and put

${\mathfrak {H'}}_{x}=\left({\frac {\partial \psi _{y}}{\partial z}}\right)'-\left({\frac {\partial \psi _{z}}{\partial y}}\right)'{\text{, etc.}}$

(37)

Once ${\mathfrak {H}}'$ is found by that, equation (III_b) provides us with the value of ${\dot {\mathfrak {d}}}$ and thus also, as far as we don't use additive constants, the value of ${\mathfrak {d}}$. From (VI_b) it also follows ${\mathfrak {H}}$; from (V_b) and (VII_b) it follows ${\mathfrak {F}}$ and ${\mathfrak {E}}$. That in this way really all the equations are satisfied, can be proven, but should not be discussed here for brevity.

In contrast, in the next section the value of $\psi _{x}$ shall be given, and in § 33 the solution for a special case shall be further developed.

It should also be remarked before, that the variable $t'$ can be regarded as a time, counting from an instant that depends on the location of the point. We can therefore call this variable the local time of this point, in contrast to the general time t. The transition from one time to another is provided by equation (34).

§ 32. The product $\rho {\mathfrak {v}}_{x}$ in the first of equations (36), as noted already, is a known function of x, y, z and $t'$. We accordingly set