A RIGID BODY IN ELLIPTIC SPACE. 
283 
Mr. Homersham Cox, “ Homogeneous Coordinates in Imaginary Geometry, and 
their application to Systems of Forces,” Quarterly Journal, vol. 18. 
For the coordinates of a line see the paper by Professor Cayley, Camb. Phil. 
Trans., vol. xi. 
On the geometry of elliptic space. 
§ 1. Geometrical theorems are sometimes divided into two classes, descriptive and 
metrical. Descriptive theorems have reference to the relative positions of figures, and 
are unaltered by projection and linear transformation. Metrical theorems have refer¬ 
ence to magnitudes, such as lengths of lines, the measures of angles, areas and volumes. 
But it has been pointed out by Professor Cayley that metrical theorems may always 
be stated as descriptive ; they are descriptive relations between geometrical figures 
and certain fixed geometrical forms, which he calls the Absolute. In ordinary 
plane geometry the Absolute consists of an imaginary point-pair on a real line, viz., 
the circular points at infinity. The magnitude of the angle between two lines, for 
instance, may be expressed as a function of the anharmonic ratio of the pencil formed 
by the lines, and the pair of lines drawn from their intersection to the Absolute point- 
pair. In three dimensions the Absolute is the imaginary circle at infinity. 
2. Professor Cayley generalises this idea of metrical theorems by supposing the 
Absolute to be the points and planes of a fixed quadric surface in space. The Absolute 
in any plane consists of the points and lines of a fixed conic lying in the plane, the 
conic being the intersection of the plane with the Absolute quadric. 
There are three different kinds of geometry of space depending on the nature of this 
Absolute quadric. These are 
(1.) Elliptic geometry, in which all the elements of the Absolute are imaginary. 
(2.) Hyperbolic geometry, in which the Absolute surface is real, but contains no 
real straight lines, and surrounds us. 
(3.) Parabolic geometry, in which the Absolute degenerates into an imaginary conic 
in a real plane. 
In what follows we shall suppose all the elements of the Absolute imaginary. 
3. On any line there is an Absolute point-pair, viz., the intersections of the line 
with the Absolute quadric. The position of any point on the line will be determined 
when we know the ratio of its distances from the Absolute points. If we denote this 
ratio by z, the distance between two points must be a function of the ratios z 1 and z 2 , 
corresponding to the points. 
Now, the fundamental property of the distance between two points may be expressed 
by the relation 
PQ+QB=PR 
where P, Q, P t are three points on the same line. In view of this relation the distance 
between two points z l5 z. 2 is defined to be 
2 o 2 
