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$V^{2}\Delta {\mathfrak {H'}}_{x}-\left({\frac {\partial ^{2}{\mathfrak {H'}}_{x}}{\partial t^{2}}}\right)_{1}=4\pi V^{2}\left\{{\frac {\partial (\rho {\mathfrak {v}}_{y})}{\partial z}}-{\frac {\partial (\rho {\mathfrak {v}}_{z})}{\partial y}}\right\}+$

$+4\pi {\mathfrak {p}}_{z}\left\{{\frac {\partial (\rho {\mathfrak {v}}_{y})}{\partial t}}-{\mathfrak {p}}_{x}{\frac {\partial (\rho {\mathfrak {v}}_{y})}{\partial x}}-{\mathfrak {p}}_{y}{\frac {\partial (\rho {\mathfrak {v}}_{y})}{\partial y}}-{\mathfrak {p}}_{z}{\frac {\partial (\rho {\mathfrak {v}}_{y})}{\partial z}}\right\}-$

$-4\pi {\mathfrak {p}}_{y}\left\{{\frac {\partial (\rho {\mathfrak {v}}_{z})}{\partial t}}-{\mathfrak {p}}_{x}{\frac {\partial (\rho {\mathfrak {v}}_{z})}{\partial x}}-{\mathfrak {p}}_{y}{\frac {\partial (\rho {\mathfrak {v}}_{z})}{\partial y}}-{\mathfrak {p}}_{z}{\frac {\partial (\rho {\mathfrak {v}}_{z})}{\partial z}}\right\}.$

§ 31. In the following calculation, magnitudes of order ${\mathfrak {p}}^{2}/V^{2}$ should be neglected. First, we neglect on the right-hand side the terms with two factors ${\mathfrak {p}}_{x}$, ${\mathfrak {p}}_{y}$ or ${\mathfrak {p}}_{z}$, since we find a similar term in $V^{2}$; and we therefore retain only

$4\pi V^{2}\left\{{\frac {\partial (\rho {\mathfrak {v}}_{y})}{\partial z}}-{\frac {\partial (\rho {\mathfrak {v}}_{z})}{\partial y}}\right\}+4\pi \left\{{\mathfrak {p}}_{z}{\frac {\partial (\rho {\mathfrak {v}}_{y})}{\partial t}}-{\mathfrak {p}}_{y}{\frac {\partial (\rho {\mathfrak {v}}_{z})}{\partial t}}\right\}.$

Second, we write for the operation that has to be applied to ${\mathfrak {H'}}_{x}$,

$V^{2}\Delta -\left({\frac {\partial }{\partial t}}-{\mathfrak {p}}_{x}{\frac {\partial }{\partial x}}-{\mathfrak {p}}_{y}{\frac {\partial }{\partial y}}-{\mathfrak {p}}_{z}{\frac {\partial }{\partial z}}\right)^{2}=\left(V^{2}{\frac {\partial ^{2}}{\partial x^{2}}}+2{\mathfrak {p}}_{x}{\frac {\partial ^{2}}{\partial x\ \partial t}}\right)+$

$+\left(V^{2}{\frac {\partial ^{2}}{\partial y^{2}}}+2{\mathfrak {p}}_{y}{\frac {\partial ^{2}}{\partial y\ \partial t}}\right)+\left(V^{2}{\frac {\partial ^{2}}{\partial z^{2}}}+2{\mathfrak {p}}_{x}{\frac {\partial ^{2}}{\partial z\ \partial t}}\right)-{\frac {\partial ^{2}}{\partial t^{2}}}=$

$=V^{2}\left({\frac {\partial }{\partial x}}+{\frac {{\mathfrak {p}}_{x}}{V^{2}}}{\frac {\partial }{\partial t}}\right)^{2}+V^{2}\left({\frac {\partial }{\partial y}}+{\frac {{\mathfrak {p}}_{y}}{V^{2}}}{\frac {\partial }{\partial t}}\right)^{2}+$

$+V^{2}\left({\frac {\partial }{\partial z}}+{\frac {{\mathfrak {p}}_{z}}{V^{2}}}{\frac {\partial }{\partial t}}\right)^{2}-{\frac {\partial ^{2}}{\partial t^{2}}}.$

The form of this expression suggests the introduction of a new independent variable instead of t

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

(34)

and to consider ${\mathfrak {H'}}_{x}$, as well as $\rho {\mathfrak {v}}_{y}$ and $\rho {\mathfrak {v}}_{z}$, as functions of x, y, z and $t'$. We denote the derivative that corresponds to this view by

$\left({\frac {\partial }{\partial x}}\right)',\ \left({\frac {\partial }{\partial y}}\right)',\ \left({\frac {\partial }{\partial z}}\right)'{\text{ and }}{\frac {\partial }{\partial t'}}$

and give to the sign $\Delta '$ the meaning

$\left({\frac {\partial ^{2}}{\partial x^{2}}}\right)'+\left({\frac {\partial ^{2}}{\partial y^{2}}}\right)'+\left({\frac {\partial ^{2}}{\partial z^{2}}}\right)'$