288 G. H. KNIBBS. 



p itself being a linear constant characteristic of the space, and 

 analogous to the radius of curvature of a surface. This briefly 

 indicates the chief properties of the two types of curved space. 



21. Geometrical illustration of elliptic and hyperbolic space. — 

 Suppose in Fig. 16, PRQS denote a sphere (i.e. a surface of con- 

 stant positive curvature) whose centre is 0. The shortest 

 distances (geodesies) on its surface will be parts of great circles 

 not greater than a semicircle ; and are the analogues, on the sur- 

 face, of straight lines on a plane. 1 



If the line EDD'F be equidistant from the great circle RAA'S, 2 

 it does not define the shortest distant between the several points: 

 these would be parts of great circles indicated by the dotted lines 

 from point to point, and would not form one and the same geodesic, 

 i.e. the angles EDQ, QDD', etc., are less than right angles, if 

 reckoned between the geodesies, and the line DQ etc., but are 

 right angles only if reckoned from the equidistant line, Q being 

 the pole of RAS. If R be the pole of PADQ, then the angles 

 RAD, RDA, EDA are all right angles; hence there cannot be 

 parallel or equidistant geodesies : that is, no figure on a surface of 

 constant positive curvature can have its opposite sides equal and 

 parallel and its four angles right angles. From the figure ADD' A, 

 taking the dotted line DD', it is easy to see that the sum of the 

 internal angles exceeds four right-angles, and that of the angles of 

 the triangle ABO, exceeds two right angles. This excess is always 

 A A 



<=W = 7 (24a) 



where A is the area of the figure and p and p" are the principal 

 radii or curvature 3 at right angles to one another and p 2 is equal 

 to their product, or in the sphere is the square of the radius. 



1 A surface (the rectifying developable) can be drawn through any 

 geodesic, in such a way that when it, the surface, is flattened into a plane 

 (developed), the curve is a straight line. 



2 Like a parallel of latitude to the Equator on a map of the Earth. 



3 Or the product of the radii of curvature in any two directions at right 

 angles to one another. 



