156 THE QUANTUM [xi . 4 
mean a two-dimensional continuum in this hyperspace, so that 
a surface is defined by two equations between x, y, z and t. 
The generalized tubes of force are surfaces defined in this 
way, and as it can be shown that they are independent of the 
arbitrary choice of a particular observer, they may be regarded 
as really objective. Whittaker first considers a region of free 
space with solitary electrons dispersed in it, and for simplicity 
assumes that the electric and magnetic vectors are everywhere at 
right angles to each other. He then shows that it is possible 
to derive a doubly infinite family of surfaces, which are covariant 
and are called the electropotential surfaces. These surfaces 
are represented by a pair of integral equations 
(f>{x,y, z, t) — a, y)(x,y, z,t) —b . . . n : 2 
where a and b are arbitrary parameters. It is because there 
are two arbitrary parameters in the equations of one of these 
surfaces that we speak of a doubly infinite family. These electro 
potential surfaces which exist in the four-dimensional world of 
space-time possess properties analogous to those of equipotential 
surfaces in electrostatics. In fact, when the field is static, each 
electropotential surface is wholly contained in three-dimensional 
space, and is an ordinary equipotential surface. By putting 
t = t Q we may determine the intersection of an electropotential 
surface with the three-dimensional “ space ” as viewed by an 
observer at any instant t Q . It then appears that “ the inter 
sections of the electropotential surfaces with the instantaneous 
space of an observer are the lines of magnetic force (in Faraday’s 
sense) of that observer at that instant. In fact, each electro 
potential surface may be regarded as a single moving Faraday 
line of magnetic force. This applies to any observer since the 
electropotential surfaces are covariant: and therefore we see 
that the electropotential surfaces may be regarded as built up 
of the Faraday lines of magnetic force of the field, as perceived 
by different observers moving in all possible directions with all 
possible velocities.” 
In the same way another doubly infinite family of surfaces 
may be shown to exist in the electromagnetic field, which may 
be called magnetopotential surfaces. These surfaces reduce to 
the ordinary equipotential surfaces when the field is purely 
magnetostatic. 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. 
Thus in four-dimensional space the electropotential surfaces 
and the magnetopotential surfaces are two covariant families of 
surfaces, and it can be shown that they are everywhere absolutely 
orthogonal.