284 . G. H. KNIBBS. 



of the same order, or a surface whose radius of curvature is an 

 infinity of higher order. The graphs of the two cases are pseudo- 

 continuously represented in (a) and (b) respectively of Fig. 15, 

 the ratio being represented on the y axes corresponding to any 

 value of x. 



The regions of impossibility indicate that the limitation of the 

 definition of distance is identical with that which requires that we 

 shall regard the operation of taking the logarithm of a negative 

 number, as an impossible one. 1 



A curious result is that the 'distance' hi is infinitesimally near 

 zero whatever the actual magnitude, if a and b be minus and plus 

 infinity from h and I, or from I and h, since it is 



2> kl = logai = log(l-^etc,)=-0 n (22) 



d denoting the length hi. 



The points a and b need not be real, but the case does not call 

 for special consideration as regards the principle of continuity. 



17, The theory of linear intensity. — It has already been noticed 

 that the elements of projection lines are not necessarily uniform 

 in their intensity. Consider in Fig. 14, the successive projection 

 of the points r, q, p, a, n, b, it is obvious that no difficulty will be 

 introduced by assuming, in tracing the correspondence from minus 

 infinity (R) to plus infinity (B), that the progression commences 

 at r and ends at b, the same point. This is of course purely 

 schematic, for no consistent antipodal scheme of representation 

 can be developed, (see Figs. 2, 9, 10, 11. 12). Nevertheless 

 algebraically it will give consistent results for the intensity of the 

 projected line. Defining, as before, intensity as the ratio of the 

 real to the projected length, i.e. r=8S/&s, see Fig. 13, it may easily 

 be shewn that if PS=yj &b=g the angle of intersection BLb = <f> t 

 and NS6 = 0, then the elements being infinitesimal, we shall have 



/ = ^ = / sin 2 (<ft + fl) (93) 



ds g ' sin 2 6 V ; 



1 It is easy to see that by changing the sign, or reforming the range, 

 J) k i may have a real value. 



