PKINCIPLE OF CONTINUITY IN THE THEORY OF SPACE. 291 



23. Impossibility of elliptic or hyperbolic space existing in a 

 pure 1 homaloid of the same number of dimensions. — We have seen 

 that a symmetrical 2-dimensional elliptic closed space can exist in 

 a homaloid of three dimensions, and that although a 2-dimensional 

 hyperbolic space can be developed, it cannot possibly, if finite, be 

 completely symmetrical, and in any case has singularities. 2 We 

 proceed to shew that neither space can possibly exist in a homaloid 

 of the same number of dimensions. 



It is unnecessary to further discuss the impossibility in 2- 

 dimensional space. Since 3-dimensional figures, in all three types 

 of space, are identical when infinitesimal, the solid angle at every 



dy/dx = tan <j> — ^[ ) ; ds/dx = sec <£ = Vf — ] ; 



hence putting b for the distance CO, we get 



p=2S{2a(2a-x)}; o- = 2V(2ax) 



Hence 



pp — (6 + x) 4a; var — (b + 4a - x) 4a 

 Consequently if b be infinite, while a is finite, or if b be an infinity of the 

 (n + l)th order while a is an infinity of the nth order only, the eycloidal 

 curve and its evolute (and identical cycloid) satisfy the conditions that 

 the radius of curvature V(pp) = </(vcr) shall be constant throughout, and 

 further that for equal distances from M toward K, and K toward O, the 

 radii of curvature in and at right angles to the meridian shall be individu- 

 ally equal. In making pp = va; we must abandon the condition that 

 the curve MQK shall be identical in shape with KPO if 6 be finite, for 

 the parameters of the two curves necessarily differ, owing to the different 

 distances from the axis of the tore ; which establishes the proposition. 

 It may be noticed that meridian lines diverge continuously in .passing 

 from the internal to the external equator. It is worthy of remark that 

 there is a very characteristic difference between the surface developed by 

 rotating KOL about ECS and an hyperboloid of one sheet ; that is the 

 eycloidal surface cannot, and the hyperboloidal can be generated by the 

 motion of a straight line. Again for the same curvature at O, the centre 

 of curvature moves from M towards the curve in the case of the cycloid 

 and away from the curve in the case of the hyperboloid. A straight line 

 cannot of course lie on a surface of constant positive or negative curvature. 

 It may be added that a solid of constant negative curvature would be of 

 the form shewn in Fig. 18a, the curves not differing greatly from cycloids: 

 the curves are obviously not identical. 



1 That is an homogeneous and isotropic homaloid. 



2 One may say that O is the position at the equator of the surface LOK, 

 and K and L, of the surfaces KML in other respects the latitude charac- 

 teristics are inverted. The singularities at KL and M, show that the 

 surface cannot be continuous in the proper sense of the term. 



