A RIGID BODY IN ELLIPTIC SPACE. 
287 
is applicable to this case also, therefore 
cos 2 0— 
( x \ x i ~t ~jjihY 
and therefore 
Hence finally 
(V+yiW+y. 8 )’ 
^ 2 +.m = s in a s i n & cos C 
cos c = cos a cos 6+sin a sin b cos C. 
From this equation may be deduced, as in Spherical Trigonometry, the relations 
sin A_sin B sin C 
sin a sin b sin c 
Hence the geometry of any plane in elliptic space is the same as the geometry of a 
sphere in ordinary space. A straight line in the plane corresponds to a great circle 
on the sphere. But further, the distance between any two points measured in the way 
we have indicated is periodic, the length of a complete period being 27 t. Hence we 
infer that the radius of any great circle of the sphere is unity. Thus any line and 
any plane may be supposed to have a uniform positive curvature unity. 
Biemann, in his memoir “ On the Hypotheses which lie at the Bases of Geometry,” 
speaks of the curvature of an w-fold extent at a given point and in a given surface 
direction; he explains it as follows :— 
Suppose that from any given point the system of shortest lines going out from it 
be constructed. Any one of these geodesics is entirely determined when its initial 
direction is given. Accordingly we obtain a determinate surface if we prolong all the 
geodesics proceeding from the given point and lying initially in the given surface 
direction ; this surface has at the given point a definite curvature, measured in the 
manner indicated by Gauss. This curvature is the curvature of the w-fold continuum 
at the given point in the given surface direction. 
If we construct a surface at a given point of elliptic space in any direction in the 
way thus indicated, the geometry of such a surface is the same as that of a sphere 
of unit radius in ordinary space. Thus for all points and for all surface directions 
the curvature will be unity. Hence elliptic space is said to have a uniform positive 
curvature. 
10. The general equation of a conic, in the notation of ordinary space, is a homo¬ 
geneous equation of the second degree. Hence, when we pass to the new coordinates, 
the equation to a conic will still be homogeneous and of the second degree. If we 
choose our triangle of reference to be the self-conjugate triangle common to the conic 
and the Absolute, the form of the equation becomes 
Ax 2 B f + Cz~ — 0. 
