(570 ) 
8. Are there not also non-linear systems Sy; and Syn of 
spherical spaces Sp, respectively K-fold and n——1-fold infinite 
situated in JS, in such a way that each spherical space of one system 
touches all the spherical spaces of the other? 
This question must be answered in the affirmative as we shall 
prove here analytically. 
If in a space S, of S, the spaces St and Sj, which have but 
the point O in common are perpendicular to each other in this point, 
if OP is the normal in OV on Sa, OQ an arbitrary line through 
O in S&, OR an arbitrary line through O in S,—,—1 and if we assume 
(fig. 3) in the planes OPQ and OPR an ellipse (/) with the half 
axes OA—=a, OB=b6 and an hyperbola (#7) with the half axes 
OC=c=Va'—b?, OD=6, then by rotation of (#) round OP 
in the space Sy — (OP, Sj) — when every point describes a spherical 
: : 3 0) 
space Sp a quadratic space of revolution Qh is generated, by 
rotation of (H) round OP in the space Sr == (OP, S, ri) — 
when every point describes a spherical space Sp: — a quadratic 
: Oe ie 
space of revolution qe, is generated. 
Fig. 3. 
If now FE and # are arbitrarily chosen points of those figures of 
revolution the distance HH can easily be calculated. If namely we 
use a rectangular system of coordinates with O as origin, OP as 
