3 
1921-22.] On Tubes of Electromagnetic Force. 
The ordinary Faraday tubes not only furnish us with a graphical 
representation of the state of the field, but they also enable us, by com- 
paring two regions at a distance from one another along the same tube, 
to establish direct connections between the fields in these distant regions : 
they enable us, in fact, to integrate the differential equations of the 
electrostatic field in an intuitive geometrical fashion. Similarly, the cala- 
moids not only provide a graphical representation of the state and history 
of the whole field, but they also enable us to integrate the general differential 
equations of the field (the Maxwell-Lorentz equations) in an intuitive 
geometrical fashion. 
§ 2. The Electropotential Surfaces. 
We shall, as the fundamental case, study a field free from ordinary 
ponderable matter, so that we have to consider only a region of free space 
with solitary electrons dispersed in it. (The formulae for the more general 
case in which ponderable matter is present may be derived from the 
formulae for this fundamental case, by supposing that each molecule of a 
ponderable dielectric contains an electric doublet, etc.) Since the electrons 
are singularities of the tubes of force, we shall for the present not consider 
the immediate neighbourhood of electrons, but shall investigate the tubes 
of force in free space, where their behaviour is regular. Moreover, we shall 
suppose the units so chosen that the velocity of light is unity. If, then, the 
three components of the electric vector are denoted by {d^, dy, d^), and the 
three components of the magnetic vector by (ha;, hy, hg), we have the usual 
Maxwellian equations of the electromagnetic field 
dhg ■ dhy ddgf,' 
dy dz dt 
dl\^ dji/g ddy 
dz dx dt 
> ( 1 ) 
dx dy dt 
dx di/dz~ J 
ddg ddy dh^' 
dy dz dt 
ddg, ddg dhy 
dz dx dt 
r (2) 
^ _dj^\ ^ ' 
dx dy dt 
dhx dhy dhg ^ 
dx dy dz 
