PRINCIPLE OF CONTINUITY IN THE THEORY OF SPACE. 313 



figure illustrating a closed 3-dimensional space of such a type, 1 

 bounded by a surface of both positive and negative curvature : it 

 suggests a type of space of more complex variety than either 

 elliptic or hyperbolic alone. 



Returning to the theory of representation on Riemann surfaces, 

 the study of their deformation and connectivity have shewn that it 

 is possible to transform the ra-ply connected surface into a simply- 

 connected surface by a series of dissections. 2 It is not proposed 

 to discuss the connectivity or dissection of the surfaces; but it 

 may be noticed that although they can never be of the character 

 alleged, viz. planes, spherical surfaces etc., figures infinitesimally 

 approximate thereto, can be generated continuously. In Fig. 32 (a) 

 it is evident from the figure itself, that the line - oo can be 

 spirally moved so that its path will be the required surface. The 

 surface of Fig. 33 (b) can be continuously generated by the motion 

 of lines from the points A and B and the line AB itself. The 

 attempt at continuous generation will reinforce the recognition of 

 their real departure from their ideal description. 



36. Conception of n-ply extended magnitude. — Reference has 

 already been made to Riemann's" treatise on the hypotheses which 

 lie at the basis of geometry, in which he affirms that that science 

 assumes as things given both the notion of space, and the first 

 principles of constructions in space I The entire argument is 

 intelligible only if those notions be first admitted, i.e. if we are to 

 be assumed capable of distinguishing in thought between straight 

 and curved plane and spherical, and so on: but it is not intelligible; 

 consequently no reason founded on such ideas can be adduced to 



3 Let a point p' be applied at p diametrally opposite, and imagine the 

 interpenetration to be possible. The figure so ' formed ' could be closed 

 up into such a figure as 34, half of which only is shaded. Its cross section 

 is a lemniscate of varying parameter. 



2 The matter has been discussed by Casorati, Clebsch, Clifford, Hof mann, 

 Klein, Luroth, Neumann, Prym, Schlafli and others. When all the wind- 

 ing points are simple, of a Riemann's surface withp sheets, of connectivity 

 2q + 1, the surface can be so transformed that there will be a single 

 branch-line between consecutive sheets excluding the last two, between 

 which there are q + 1 branch lines. This surface is known as the canonical 

 form for the case where all branch-points are simple. 



