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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

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

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

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