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