272 G. H. KNIBBS. 



is 1 - 2co = = + 2co . That this is an essentially artificial 

 representation 2 is evident from the following considerations, viz.: 

 1° The discontinuity is resolved through an infinite path of the 

 first order, but cannot be resolved within the dimension itself. 2° 

 Though it can be resolved in space of higher dimension, even that 

 space has no unique position, since the circle of continuity may be 

 anywhere on the surface of an infinite tore, see Fig. 1. 3° The 

 curvature is zero only /or finite geometrical figures, and to the first 

 order of infinitesimal? and is consequently distinguishable from 

 the lesser curvature of linear "space" of a higher order of infinity. 



This scheme of resolving infinite discontinuity can be extended 

 to space of any dimensions by infinite unbounded figures of a 

 higher dimension. Reverting to equation (9), if we put a = 1 and 

 treat x as the independent and y as the dependent variable, we 

 obtain the following values 



x ± oo, ± 2 ; y = ± 2 , ± oo 

 that is the equation represents two straight lines apparently 4 

 crossing one another at whatever angle x makes with the y axis. 

 On a sphere of infinite radius (of the first order) and therefore of 

 (first order) zero curvature, these may be represented as in Fig. 2, 

 by the lines O'X'OY'O'YOXO'X', developed in that order. The 

 order and the resolution of the discontinuity, are better illustrated 



by the hyperbolas 



xy = \ (12), 



shewn by the heavier lines with arrows indicating the direction of 

 progression, the generation being - to + . The lines are the 

 locus P, the terminals of y, its distance from x continually satisfy- 

 ing (12). When the radius 00 is finite, the curves will lie on 

 opposite sides of the axes XOX', YOY', as shewn Fig. 2, but when 

 it is infinite, they are distant only 0y4ir from the axes; hence, 



1 2x is sometimes regarded as the limit of the even numbers. See 

 Wead, Some discontinuous and indeterminate functions. Phil. Soc. 

 Washington, Bull, xiv., p. 66, 1900. 



2 That is a purely schematic representation always distinguishable 

 from the rigorous representation. 



3 As may be seen by multiplying (11) by infinity. 



4 It will be seen that the y line does not really cross the x line. 



