12 
Proceedings of the Koyal Society of Edinburgh. [Sess. 
dimensional space corresponding to this value of V’), and therefore the 
surface S lies wholly in this hyperplane: and the property of half- 
orthogonality now implies that at every point of intersection of 2 with 
an equipotential surface, their tangent-planes are perpendicular in the 
ordinary three-dimensional sense. That is to say, 'E is a tube of force as 
defined in ordinary electrostatical theory. 
Similar reasoning applies to the case when the field is purely 
magnetostatic. 
It appears therefore that the surfaces Z are a covariant family of 
surfaces which, wdien the field is purely electrostatic or purely magneto- 
static, reduce to the ordinary Faraday tubes of force. We shall therefore 
call them the tubes of force of the electromagnetic field. It is convenient 
to introduce a new term, partly for brevity and partly in order to dis- 
tinguish them from the Faraday tubes, which are limiting cases of them. 
We shall call them calamoids.^ 
§ 8. The Parallel Properties of the Calamoids. 
We have seen (§7) that the tangent-plane to a tube of force or calamoid 
Z at P intersects the tangent-plane to the electropotential surface o- at P 
in a line, and therefore these tangent-planes have a point at infinity in 
common : that is, they are half-parallel. A similar argument applies to 
the magnetopotential surface r. Thus we have the theorem that a calamoid, 
at every one of its points, is half-parallel and half -orthogonal to the 
electropotential surface which passes through the point, and is also half- 
parallel and half -orthogonal to the magnetopotential surface which passes 
through the point. 
§ 9. The Conditions for Half-Parallelism and 
Half-Orthogonality. 
We shall now find the analytical conditions which must be satisfied 
when two planes in our space-time hyperspace are half-parallel or half- 
perpendicular. 
First, let us consider half-parallelism. 
Let one plane ^ have the equations 
( + + + 4 = 0 
I + + r]2.^ + r]y; + r]^ = Q, 
and let the other plane have the equations 
\ 7}^' X + 7]^y -1- 4- 7) ft 4- 7]l = 0. 
* From KaKafxos, a reed-pipe. 
