PBINCIPLE OF CONTINUITY IN THE THEORY OF SPACE. 311 



four-sheeted surface similar to Fig. 32 (c) and (e), O being the 

 branch point. 1 



When surfaces other than planes are required, the nature of 

 junction and passage from sheet to sheet is in all cases identical, 

 i.e. by a line; sufficient however has been indicated to shew the 

 characteristics of the scheme. Strictly it is not a realisable or 

 possible scheme, as is evident if we remember that the infinitesimal 

 is a real quantity not absolutely nothing, hence the surfaces are 

 not really planes, spheres, etc., or if they were could not join in 

 the manner required, 2 



35. The connectivity of space. — The connectivity of surfaces, of 

 solids or of n-dimensional quanta, expresses the number of sections 

 that must be made in order to divide them into two distinct i.e. 

 simply connected parts. Consider Fig. 33 (a): it will be observed 

 that its edge is continuous, and that it is unifacial, or unilateral* 

 and that starting at any point 0, a single circuit in the direction 

 of the arrow, terminates at the underside of O; a second circuit 

 however returns to the starting point. 4 (This is the nature of 

 supposed single elliptic space). Further we notice that Fig. 33 (b) 

 has only one edge, and Fig. 33 (c) but one edge and face : it is 

 therefore similar to (33a): a single circuit from O does not return 

 thereto, but ends on the under-side. 



The nature of the connectivity of surfaces and solids is by no 

 means so obvious as might at first be imagined. If for example 

 in Fig. 33 (a), representing a band of uniform width, (which has 

 been turned through ir so as to make it unifacial and single-edged, 

 instead of cylindrical) a cut be started at A and kept a uniform 



1 See Holzmxiller's Einfiihrung in die Tbeorie der isogonalen Verwand- 

 schafteti und der conformen. Abbildung, Leipzig 1882 (Teubner). The 

 dotted lines will indicate the complete path from sheet to sheet. 



2 The movement from sheet to sheet does not introduce even infini- 

 tesimal error of the first order, with regard to the purpose of the repre- 

 sentation : hence practically it is unexceptionable. 



3 If I mistake not, Mobius was the first to notice this fact. It may be 

 mentioned that it has been suggested that the ordinary plane of projective 

 geometry is unilateral ! See Klein, Math. Ann., Bd. vn., p. 549. 



4 Hence if cut along the arrow path it will not be divided. 



