9 
1921-22.] On Tubes of Electromagnetic Force. 
Comparing (6) and (7), we see that the iwojections of any surface-element 
of an electropotential surface, on the six co-ordinate planes 
xiy, xZj xt, yz, yt, zt, 
are respectively proportional to 
Thus the inclinations of the electropotential surface at any point to the 
co-ordinate planes specifies the nature of the electromagnetic field at that 
p)oint. 
(ii) Consider the intersection of an electropotential surface with the 
three-dimensional space ” observed by an observer at any instant t^. This 
instantaneous space will be the hyperplane t = tQ. The intersection of the 
hyperplane t — t^ with the electropotential surface will be a curve, and on 
putting dt — 0 in the equations (4) which define the electropotential surface 
we see that for this curve we have 
dx dy dz 
hx hy hf 
That is to say, the intersections of the electropotential surfaces with the 
instantaneous spaee of an observer are the lines of magnetic force {in 
Faraday’s sense) of that observer at that instant. In fact, each electro- 
potential surface may be regarded as a single moving Faraday line of 
magnetic force. 
This applies to any observer, since the electropotential surfaces are 
covariant: and therefore we see that the electropotential surfaces may 
be regarded as built tip of the Faraday lines of magnetic force of the field, 
as perceived by different observers moving in all yjossible directions with 
all possible velocities. 
§ 5. The Magnetopotential Surfaces. 
We shall next introduce another family of oo ^ surfaces which exist 
in the electromagnetic field. 
Consider the set of total difierential equations 
dydy - dydz — hy.dt = 0 
— dydx 4 - d^dz — liydt — 0 
dydx — dg^dy — h^dt = 0 
hy.dx + hydy + li^dz = 0 
By reasoning similar to that in § 2, we can prove that the solution 
of this set of total differential equations represents a family of go ^ 
surfaces in the four-dimensional hyperspace. These will be called the 
magnetopotential surfaces of the electromagnetic field. As in §§ 2, 4, 
