Page:Über die Möglichkeit einer elektromagnetischen Begründung der Mechanik.djvu/3

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The electric and magnetic quanta are therefore given as integration constants, when we differentiate equations (1) with respect to x, y, z and then sum up. Then it is namely

\frac{\partial}{\partial t}\left(\frac{\partial X}{\partial x}+\frac{\partial Y}{\partial y}+\frac{\partial Z}{\partial z}\right)=0,\quad\frac{\partial}{\partial t}\left(\frac{\partial L}{\partial x}+\frac{\partial M}{\partial y}+\frac{\partial N}{\partial z}\right)=0,

thus

(2) \frac{\partial X}{\partial x}+\frac{\partial Y}{\partial y}+\frac{\partial Z}{\partial z}=-4\pi\varsigma,\quad\frac{\partial L}{\partial x}+\frac{\partial M}{\partial y}+\frac{\partial N}{\partial z}=-4\pi m,

where \varsigma and m are independent from time, thus being temporally invariable quanta.

If one multiplies the first row of equations (1) with X/4\pi, Y/4\pi, Z/4\pi,, the second with L/4\pi, M/4\pi, N/4\pi,, and sum up all of them, one thus obtains after partial integration over a closed space, whose surface normal shall be n and surface element be dS, the theorem

(3) \begin{cases} \frac{1}{8\pi}\frac{d}{dt}\int\int\int dx\ dy\ dz\ \left(X^{2}+Y^{2}+Z^{2}+L^{2}+M^{2}+N^{2}\right)\\ =\int dS\left[\left(YN-ZM\right)\cos(xn)+(ZL-XN)\cos(ny)+(XM-YL)\cos(nz)\right].\end{cases}

If either X, Y, Z or L, M, N vanish at the surface, we thus have

(4) \frac{1}{8\pi}\int\int\int dx\ dy\ dz\ \left(X^{2}+Y^{2}+Z^{2}+L^{2}+M^{2}+N^{2}\right)=const.

Those on the left, that are always being constant when summed over a sufficiently great space, we denote as the electromagnetic energy.

Now we make the assumption, that the mechanical processes are of electromagnetic nature as well, i.e., that they can be developed from the foundations considered.

For that purpose, we assume at first, that the substrate denoted as matter is composed of positive and negative electric quanta, namely of such elementary quanta,