10 
Proceedings of the Poyal Society of Edinburgh. [Sess. 
we can show that the magnetopotential surfaces are a covariant family 
of surfaces, which exist in electromagnetic fields, and which reduce to 
the ordinary equipotential surfaces when the field is purely magneto- 
static. In the general case, the intersections of the magnetopotential 
surfaces with the instantaneous space of an observer are the lines of 
electric force {in Faraday' s sense) of that observer at that instant. 
§ 6. Mutual Relations of the Electropotential and 
Magnetopotential Surfaces. 
Two surfaces in four-dimensional hyperspace intersect in general in 
solitary points (this is to be contrasted with the fact that two surfaces 
in three-dimensional space intersect in general in a curve). Suppose then 
that {x, y, z, t) is a point of intersection of any electropotential surface 
with any magnetopotential surface : let (dx-^, dy-^, dz^, dtf) be a line-element 
issuing from this point and lying in the electropotential surface, and let 
(dx^, dy. 2 , dz^, dtf) be a line-element issuing from the same point and lying 
in the magnetopotential surface. 
Then from equations (4) and (8) we have 
h^dy^ = hydzy - dy.dt^ 
li^dx^ = h^dz-^ -i- dydt-^ 
h^dt^ = dydx.2 - dxdij^ 
h^dz^ — — hxdx^ + hydy^. 
If we substitute from these equations in the expression 
hjydxpix^ + dyydy^ + dzplz^ - dtyHf) 
it vanishes identically. Similarly, we see that the same equation is true 
when any of the other components of the electric or magnetic vectors is 
put in place of hz : and therefore we have 
dxydx^ - 1 - dyydy^ + dzylz^ — dtydt^ = 0 . 
But this is the condition that the line-elements (dx-^, dy^, dz-^, dtf) and 
{dx^, dy^, dz^, dtf) should be orthogonal : and thus we have the following 
theorem: At the point of intersection of any electropotential surface with 
any magnetoyootential surface, every line drawn in the tangent-plane 
to the one surface is orthogonal to every line drawn in the tangent-plane 
to the other surface: or, in other words, at the point of intersection of any 
electropotential surface with any magnetoyjotential surface, the tangent- 
planes to the two surfaces are absolutely orthogonal, or the electropotential 
surfaces and the magnetopotential surfaces are two families of surfaces 
ivhich are everywhere absolutely orthogonal. 
