PRINCIPLE OF CONTINUITY IN THE THEORY OF SPACE. 



273 



rejecting any consideration of infinitesimals of the second order, 

 both curves may be supposed to cross the axes at the point. In this 

 view, the differences between + and - , may be so construed as 

 to lose their significance when applied to infinity, and by this 

 scheme four infinite discontinuities disappear and disclose a single 

 pseudo-continuous curve, which in relation to other finite figures, 

 at finite distances from the origin, fulfils the requirements of 

 ordinary physical interpretations. There is of course no limit to 

 the resolution of discontinuities of this type. For example the 

 curve 



y 



± v 



( x(x — l)(x — 2 

 1 x + 3 



.(13) 



has six branches, represented in Fig. 3, forming on the infinite 

 sphere a (pseudo) continuous line. On a spherical surface the 

 discontinuity of the small oval figure — see Fig. 3a — from x = to 

 85=1, remains unresolved. This however is resolvable on a 

 different surface as will now be shewn. 



A curve of the type 



xy 2 =l (14) 



y being the dependent variable, gives four hyperbolic curves, two 



R— Dec. 4, 1901. 



