2 
Proceedings of the Royal Society of Edinburgh. [Sess. 
of equality, whatever be their velocities. But if we take two observers 
who are at any instant at the same point, but who are moving with 
different velocities, their measures of the electric and magnetic forces 
will not be concordant : and therefore it is not possible to construct a 
system of Faraday tubes which is purely objective, and independent of 
the arbitrary choice of an observer. A system of tubes of force which is 
to be really objective {i.e. depending only on the field and not at all on 
the peculiarities of the person observing it) can exist only in the four- 
dimensional world of space-time with which the Theory of Relativity has 
made us familiar. This is the standpoint of the present paper. 
The surfaces here introduced, to which I give the name calamoids, 
may be regarded as a direct generalisation of Faraday’s tubes, since they 
reduce to Faraday’s electric tubes when the held is purely electrostatic, 
and they reduce to Faraday’s magnetic tubes when the field is purely 
magnetostatic: and between these extreme types they provide a con- 
tinuous transition. The way in which the generalisation is made may 
be illustrated by the following statement: With Faraday’s electrostatic 
tubes, as we proceed along a tube, the magnitude of the electric force at 
any point is inversely proportional to the area of the cross-section of the 
tube (the cross-section being part of an equipotential surface), and the 
three components of the electric force are to each other in the same ratios 
as the areas of the projections of this cross-section on the three co-ordinate 
planes. With the calamoids introduced in the present paper, the quantity 
{(Electric Force)^ - (Magnetic Force)^} 
(which, as is well known, is covariant with respect to all Lorentz trans- 
formations) is inversely proportional to the area of the cross-section of 
the calamoid (the cross-section being part of what is here called an 
electropotential surface), and the six components of the electric and 
magnetic forces are connected in a simple way with the areas of the six 
projections of this cross-section on the six co-ordinate planes of xy, yz, zx, 
xt, yt, zt. The calamoids are covariant with respect to the Lorentz trans- 
formations — which implies that they are the same, whatever be the 
observer whose measures of electric and magnetic force are used in 
constructing them. 
The circumstance that the two kinds of Faraday tubes (electric and 
magnetic) are particular cases of calamoids seems to terminate the rivalry, 
so to speak, which has existed between them, and which has manifested 
itself in such questions as whether both of them should be regarded as 
physically existent, or only one kind, and if so, which kind. 
