PRINCIPLE OF CONTINUITY IN THE THEORY OF SPACE. 269 



Values of a 



= +o l 







+ 2 





+o n 



Value of x or y. 









Values of y 



or x. 





oo 11 



tO d+1 







T n + 2 





.. t0 2d 



00 . 



02 







03 





.. n +i 



1 



0i 







02 





.. 0" 



Oi 



0° = 1 







01 





.. 0*-* 



02 



0-1 = 



CO 





0° = 1 





.. 0»- 2 



0* 



Q-n + 1 



= 



O0 n " 



-1 Q— n + 2 = 



-- oo n ~2 



... 0° = 



(10) 



Passing now to a geometrical interpretation, if x and y are vectors, 

 i.e. quantities parallel to two axes, w T hether orthogonal or inclined 

 is immaterial, 1 we may observe that if (9) be understood as defin- 

 ing the dependence of y upon x and a, i.e. if it signify that y = ajx, 

 then, the path P, traced by one terminal of the line y as its other 

 terminal moves continuously along the aj-axis from — oo to + go, 

 departs from that axis amounts which depend upon the orders of 

 infinity and zero. Thus if a = Q 2 , which is the normal form for 

 x = 0\ y = 1 ) it will be noticed that y = l for x = 1 ; while for 

 x = 2 , 3 , y= 1, oo. Similarly if a be infinitely greater than 2 , i.e. 

 if it be 1 , then the value of y is a first order infinitesimal from 

 x= -lto#= -(Q l + 8x) and at 1 becomes — 1. Forsc^+0 1 , 

 2/=+l, and for + (0 1 + 8x) a first order infinitesimal as far as 

 as= +1, reducing to a second order infinitesimal for x = oo. If 

 conceptually we slur over the passing from - 1 to +0 1 as insig- 

 nificant, we lose sight of the extension of the discontinuity from 

 — 1, + 1, to - oo", + oo n . The defect in continuity of a discrete 

 point geometry, leaves therefore certain possibilities of interpre- 

 tation unrevealed. If a = 2 be multiplied by oo 2 , it will become 

 1, which is equivalent to enlarging the scale infinitely in both 

 dimensions. The graph then becomes two opposite hyperbolas, 

 equilateral if the axes are orthogonal. The results for different 

 values of x from plus to minus infinity of the first order, and 

 slurring over the first order infinitesimals - 1 to + 1 , are 



y= -0 1 - 1 - oo + oo +1 +0 1 

 that is the discontinuity is from minus to plus infinity. If how- 



1 The axis may also be curved or straight, provided " parallel " is 

 suitably defined. 



