6 
Proceedings of the Koyal Society of Edinburgh. [Sess. 
But the latter equations define the ordinary equipotential surfaces of the 
electrostatic field : and thus we see that the electropoteritial surfaces are a 
Govariant family of surfaces, which exist in electromagnetic fields and 
which become the ordinary equipotential surfaces when the field is purely 
electrostatic. 
§ 3. The Absolute. 
In the course of this paper there will often be occasion to use terms 
belonging to metrical geometry, such as distance, parallel, perpendicular. 
It may be well at this stage to explain precisely what is meant by them. 
The metrical properties of any kind of space are determined by what is 
called the absolute of that space. In the ordinary Euclidean geometry of 
the plane the absolute consists of a pair of imaginary points, namely, the 
'' circular points at infinity ” ; all metrical properties can be defined in 
terms of these points, e.g. if 01, OJ are the lines drawn from a point 0 
to the circular points at infinity, then two lines OA, OB through O are 
perpendicular when the lines OA, OB, 01, OJ form a harmonic pencil. 
If (x, y ) are ordinary rectangular co-ordinates in the plane, and if we write 
so that (x^, X 2 , xf) are the homogeneous co-ordinates of a point, then the 
two circular points at infinity are represented by the equations 
xf -hx^^ = 0, £Cg = 0. 
Similarly, in Euclidean geometry of three dimensions the absolute on which 
all metrical properties depend is an imaginary circle at infinity, represented 
by the equations (in homogeneous co-ordinates) 
xf + x.2^ + xf = 0, x^ = 0, 
and in the Euclidean geometry of four dimensions the absolute is an 
imaginary sphere at infinity represented by the equations 
xf + xf + xf -hxf = 0, x^ = 0. 
The four-dimensional hyperspace with which we are concerned in the 
present paper is formed of the aggregate of all the three-dimensional 
“ instantaneous spaces ” perceived by an observer at successive instants. 
This aggregate is the same for all observers, although from the Theory of 
Relativity we know that its dissection into instantaneous spaces is different 
for different observers. This hyperspace is not Euclidean, because the time t 
is different in character from the three space co-ordinates x, y, z. Writing 
X-. 
2 = ^, t = 
