7 
1921-22.] On Tubes of Electromagnetic Force. 
so that {x-^, x^, x^, x^) are homogeneous co-ordinates, we can express 
the difference in character between the time co-ordinate and the space 
co-ordinate by taking the absolute to be 
-1- X2^ -f - c^x^‘^ = 0, x^ = 0, 
where c denotes the velocity of light. ^ For simplicity, we choose our 
units so that c = l, and the absolute will therefore be taken to be 
Xj^ X2^ + - x^^ = 0, x^ = 0 . 
The negative sign which affects x^^ is the analytical way of expressing the 
difference between time and space. 
The transformations, which are called Lorentz transformations ” in 
electromagnetic theory, transform this absolute into itself : and the 
“ restricted theory of relativity is nothing but the invariant theory of the 
four-dimensional world of space and time with respect to the transformations 
of this group. 
With this absolute, geometry is non-Euclidean and of the ‘‘hyperbolic” 
type : the distance between two points whose co-ordinates are {x, y, z, t) 
and {x\ y', z\ t') is 
- {y - y'f ^y +{t- ty}- 
The area of a region on a surface which is defined by the equations 
is 
x = x{u,v), y = y{u,v), z = z{u,v), t = t{u,v) 
II 
/ 0(^1 2 
_ l0(w, v)f 
f d(x, z) Y 
\0(Z^, V)) 
-t- 
/ %> _ f Hv, ^) V 
l0(?^, v)J \d(tl, v)i l0(?^, v)j 
/0(.,O\“~ 
l0(w, V)i _ 
I 
du dv. 
Let us now introduce the notion of parallelism. Consider two planes. 
Each plane meets the hyperplane -j* at infinity, 0)5 = 0, in a line. If these 
lines do not intersect, the planes are said to be not parallel. If the lines 
intersect in a point, the planes are said to be half-parallel. If the lines are 
coincident, the planes are said to be absolutely, parallel. Thus in four- 
dimensional hyperspace there are two kinds of parallelism. 
Next, consider perpendicularity. Two lines are said to be orthogonal or 
perpendicular when the points in which they meet the hyperplane at 
infinity are conjugate with respect to the absolute. With regard to planes, 
there are two possible degrees of orthogonality, just as there are two 
degrees of parallelism ; two planes may be either — 
So far as I know, this remark was first made by Klein, 
t The locus of points whose co-ordinates satisfy a linear equation 
ax + hy + cz + dt + e = Q 
is called a hyperplane. The intersection of two hyperplanes is in general a plane, and the 
intersection of three hyperplanes is in general a straight line. 
