5 
1921-22.] On Tubes of Electromagnetic Force. 
On applying this criterion to any two of the equations (4), we find that 
these conditions of integrability are satisfied, by virtue of the equations (1), 
(2), (3) : and therefore the solution of the total differential equations (4) is 
represented by a pair of integral equations 
<fi(x,y,z,t) = a, xlf{x,tj,z,t) = h .... (5) 
where a and h denote constants of integration. 
The functions and ip are solutions of the set of partial differential equations 
“adjoint” to the set of total differential equations: these adjoint equations in our 
case are any two of the equations 
-•I 
h XI, xi V -0 
Let us now express our result in the language of geometry. Let us use 
the word surface to mean a two-dimensional continuum in the four- 
dimensional hyperspace in which x, y, z, t are co-ordinates, so that a 
surface is defined by two equations between x, y, z, t. Then evidently we 
have proved that the solution of the set of total differential equations (4) 
represents a family of cc^ surfaces in the four-dimensional hyperspace in 
which {x, y, 0 , t) are the co-ordinates. These will be called the electro- 
potential surfaces of the electromagnetic field. 
The statement that there are 00 ^ surfaces means that there are two arbitrary 
parameters in the equation of one of these surfaces, namely, the a and h of 
equations (5). 
In order to understand the nature of these electropotential surfaces, let 
us consider for a moment the particular case of a purely electrostatic field, 
for which the magnetic vector everywhere vanishes and the components 
of the electric vector are the derivates of an electrostatic potential V 
{x, y, z ) : thus, 
/?a, = 0, hy = 0, 7^2 = 0, 
® 0a; 
7 9^ 
dy = — 
dy 
dv 
dz 
The equations (4) now reduce to the following : 
r dt = 0 
Jav, 0Y, 0V, . 
of which the integrals are 
j t= Constant 
iV(a:, y, z) = Constant. 
