1921-22.] On Tubes of Electromagnetic Force. 
11 
§ 7. Intkoduction of the Tubes of Electromagnetic Force. 
Now let any continuous set of oo ^ electropotential surfaces be chosen 
arbitrarily and grouped together. Their aggregate is a three-dimensional 
region or hypersurface, which we shall call G. Moreover, let any con- 
tinuous set of 00 ^ magnetopotential surfaces be chosen arbitrarily and 
grouped together. Their aggregate is another hypersurface, which we 
shall call H. The hypersurfaces G and H will intersect in a surface; 
let us call this surface 2. We shall now investigate its properties. 
Let P be a point on 2 ; let o- be the electropotential surface through 
P, and let r be the magnetopotential surface through P. Since 2 and cr 
are two surfaces both contained in the same hypersurface G, they intersect 
in a curve. (This is to be contrasted with the fact that two surfaces in 
four-dimensional hyperspace intersect in general only in solitary points.) 
The tangent-plane to 2 at P therefore intersects the tangent-plane to o- 
at P in a line. Similarly, the tangent-plane to 2 at P intersects the 
tangent-plane to t at P in a line. 
Now we have seen (§ 6) that any line in the tangent-plane to r at P 
is orthogonal to every line drawn in the tangent-plane to or at P. There- 
fore the tangent-plane to 2 at P contains one line which is orthogonal 
to every line drawn in the tangent-plane to cr at P. Hence, at P the 
tangent-plane to 2 is half-orthogonal to the tangent-plane to cr. Similarly, 
the tangent-plane to 2 is half-orthogonal to the tangent-plane to t- Thus 
we have the theorem that the surf ace 2 is, at every one of its points, lialf- 
orthogonal to the electropotential surface wliich passes tJirough the p>oint, 
and is also half -orthogonal to the magnetop)otential surface which passes 
through the point. 
We may remark that the property here called “ half -orthogonality ” is 
really the same as what in ordinary three-dimensional geometry is simply 
called the “ perpendicularity ” of two planes : for two planes are said to 
be perpendicular to each other in ordinary three-dimensional geometry 
if one of them contains a line which is perpendicular to all the lines of 
the other. 
When the field is purely electrostatic or purely magnetostatic, the 
“surfaces 2,” which have been introduced, become the ordinary Faraday 
“tubes of force.” For, taking the electrostatic case, the electropotential 
surfaces reduce, as we have seen (§ 2), to the ordinary equipotential 
surfaces. When we take as hypersurface G the aggregate of all the equi- 
potential surfaces which correspond to a fixed value of t, these equipotential 
surfaces all lie in the same hyperplane (namely, the “ instantaneous three- 
