FOUR-DIMENSIONAL TUBES 
157 
xi. 4] 
Whittaker then proceeds to show that a surface, E, can be 
found which is at every one of its points, half-orthogonal to the 
electropotential surface which passes through the point, and is also 
half-orthogonal to the magnetopotential surface which passes 
through the point. It may be remarked that the property here 
called “ half-orthogonality ” is really the same as what in ordinary 
three-dimensional geometry is simply called the “ perpendicu 
larity ” of two planes. 
When the field is purely electrostatic, or purely magnetostatic, 
the surfaces E become the ordinary Faraday tubes of force. 
In the general case the surfaces are a covariant family of surfaces, 
which may be called the tubes of force of the electromagnetic 
field or calamoids (from KaXafiog, a reed-pipe). Thus the Faraday 
electric and magnetic tubes are not distinct and rival things, but 
two limiting cases of the same thing. 
It can be shown that these four-dimensional tubes of force 
possess important properties analogous to those belonging to 
ordinary Faraday tubes. In electrostatics there is a well-known 
result, which states that “ the cross-section of a Faraday tube, 
multiplied by the value of the electric force, is constant along the 
whole length of the tube.” The corresponding theorem in the 
four-dimensional world asserts that the cross-section of a thin 
calamoid (measured by the area which it cuts off on the electro 
potential surfaces which intersect it in curves) multiplied by the 
value of \/(D 2 — H 2 ) is constant along the whole length of the 
calamoid. Here D represents the electric vector, and H the 
magnetic vector at a point. The quantity 
\/{(Electric Force) 2 — (Magnetic Force) 2 } 
is well known to be covariant with respect to all Lorentz trans 
formations. The calamoids are also covariant—which implies 
that they are the same, whatever be the. observer whose measures 
of electric and magnetic force are used in constructing them. 
It is not very easy to get a clear mental picture of the cala 
moids, especially in their most general form. Milner * has 
treated the question in a somewhat different way with the 
intention of extending Whittaker’s results and enabling the 
reader to construct a representation of electromagnetic lines in 
hyperspace. In the electromagnetic field the directions of the 
electric and magnetic forces are in general neither along the same 
line nor at right angles to each other. Observers in different 
hyperplanes will form different conceptions of the strengths and 
directions of the electric and magnetic forces. Milner shows, 
however, that when viewed in a suitable hyperplane, i.e., when 
suitably transformed, the electric and magnetic forces at any 
* S. R. Milner, Phil. Mag., vol. 44, p. 705, 1922.