8 Proceedings of the Koyal Society of Edinburgh. [Sess. 
(1) absolutely orthogonal, namely, when every line which lies in one 
plane is orthogonal to every line which lies in the other plane ; 
or 
(2) half-orthogonal, namely, when one of the planes contains .a line 
which is orthogonal to all the lines in the other plane ; 
or 
(3) not orthogonal at all. 
§ 4. Properties of the Electropotential Surfaces. 
(i) In electrostatics the equipotential surfaces have the property that 
the projections of any surface-element on the three co-ordinate planes are 
proportional to the three components of the electric vector. We shall now 
show that in the general electromagnetic field the electropotential surfaces 
have the property that the projections of any surface-element on the six 
co-ordinate planes are proportional to the six components of the electric 
and magnetic vectors. 
To prove this, we remark that if a surface is defined by the equations 
x = x{u, v), y = y{u, v), z = z{u, v), t = t(u, v), 
then (the absolute being that described in § 3) the projections of a surface- 
element on the planes of 
xy, 
xz, xt, yz. 
, yt, zt 
respectively 
are 
'b{x, dy^ dx)^ 
(in, 
dv 
, du dv, 
d{u, v) 
dfa, v) 
d{u, v) 
d(u, v) 
d{u, v) 
{i 
d{u, v) 
Now for an equipotential surface we have from the first of equations (4) 
. dy dz dt 
dy 
dz 
dt 
■dv 
dv 
dv 
whence we have at once 
hz hy d/j/, 
t) ~ d(y, t) ~ d{y, zY 
d{u, v) d{u, v) d(u, v) 
and by use of the other equations (4) we extend this so as to obtain 
y) g) 'd{x, t) d(y, z) d{y, t ) d{z, t) 
d{u, v) d{u, v) d{u, v) d{u, v) d{u, v) dfii, v) 
