Jump to content

Page:BatemanElectrodynamical.djvu/5

From Wikisource
This page has been validated.
1909.]
The transformations of the electrodynamical equations.
227

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

(II)
(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 in this space be assigned by equating x, y, z, t to one-valued differentiate functions of two parameters , and let be the boundary of a three-dimensional manifold in which the coordinates are like functions of three parameters , of which on , and on . Then any term such as may be interpreted to mean taken over , and any term such as may be interpreted to mean taken over .

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 , 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 .


  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.