Page:BatemanElectrodynamical.djvu/5

These equations may be replaced by the two integral equations^[1]

$\iint \left(H_{x}dy\ dz+H_{y}dz\ dx+H_{z}dx\ dy+E_{x}dx\ dt+E_{y}dy\ dt+E_{z}dz\ dt\right)=0,$

(II)

${\begin{array}{l}\iint \left(E_{x}dy\ dz+E_{y}dz\ dx+E_{z}dx\ dy-H_{x}dx\ dt-H_{y}dy\ dt-H_{z}dz\ dt\right)\\\qquad =-\iiint \left(\rho w_{x}dy\ dz\ dt+\rho w_{y}dz\ dx\ dt+\rho w_{z}dx\ dy\ dt-\rho dx\ dy\ dz\right),\end{array}}$

(III)

provided the integrals receive suitable interpretations. The interpretation that first suggests itself is obtained by regarding (x, y, z, t) as the coordinates of a point in a space of four dimensions. Let any closed two-dimensional manifold $S_{2}$ in this space be assigned by equating x, y, z, t to one-valued differentiate functions of two parameters $\alpha ,\beta$, and let $S_{2}$ be the boundary of a three-dimensional manifold $S_{3}$ in which the coordinates are like functions of three parameters $\alpha ,\beta ,\gamma$, of which $\gamma =0$ on $S_{2}$, and $\gamma <0$ on $S_{3}$. Then any term such as $\iint H_{x}dy\ dz$ may be interpreted to mean $\iint H_{x}{\frac {\partial (y,z)}{\partial (\alpha ,\beta )}}d\alpha \ d\beta$ taken over $S_{2}$, and any term such as $\iiint \rho \ dx\ dy\ dz$ may be interpreted to mean $\iiint \rho {\frac {\partial (x,y,z)}{\partial (\alpha ,\beta ,\gamma )}}d\alpha \ d\beta \ d\gamma$ taken over $S_{3}$.

The relations (II) and (III) may now be obtained with the aid of (I) by applying the generalized Green-Stokes theorem as given by Baker,^[2] Poincaré,^[3] and others.

In order that equations (II) and (III) may be equivalent to (I), the axes must form a right-handed system. If we wish to use left-handed axes we must change the sign of H in (II) and (III).

We shall now endeavour to give a simpler interpretation to the integrals occurring in equations (II) and (III).

Let S be an arbitrary closed surface in the (x, y, z) space, and let t be expressed in terms of x, y, z by an arbitrary law $t=t(x,y,z)$, which must be chosen, however, in such a way that t is a single-valued function which is finite together with its derivatives with regard to x, y, z at all points within S and on S itself. Let the coordinates of points on S be expressed in terms of two parameters $\alpha ,\beta$.

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1. The integral forms occurring in these equations have been studied by Hargreaves, Camb. Phil. Trans., Vol. 21, p. 107 (1908).
2. Camb. Phil. Trans., Vol. 18 (1900), p. 408.
3. Acta Math., t. 9 (1887), p. 321.