PRINCIPLE OF CONTINUITY IN THE THEORY OF SPACE. 



283 



l0g c p M + l0gc Pun = l0gc Pkm (21) 



where the logarithm may be to any base c. Hence log c p M may 

 be denned as the projective distance between any two points k and 

 I, and is determined by the place, not merely of the points them- 

 selves, but in relation to two reference points viz. a and b, which 

 therefore may be called the absolute point-pair. Since the points 

 AKL...B and akl...b are projective, and therefore equianharmonic, 

 the projective distances oj corresponding points are also identical, 

 although, as in the illustration, one of the points B is 'at infinity.' 

 Reverting now to § 15 and to formula (18), it is easy to see that 

 this proposition is only infinitesimally approximate 1 for a first 

 order infinity, but is more nearly true for a higher infinity. If 

 only one of the points k, I is between a and b the an harmonic ratio 

 is negative, hence the distance as defined is impossible. 2 If the 

 points be in the order abkl, and a be infinitely distant from b, then 

 the anharmonic ratio becomes simply the ratio of the distances 

 bl : bk that is say l/k if I and k are reckoned from b. Hence if k 

 become identical with b the anharmonic ratio is infinite unless at 

 the same time a becomes identical with b, then it is 1. If I is at 

 infinity the anharmonic ratio ^Z is the ratio of ak : bk, i.e. say k/k'. 



Consider the finite range abk, and suppose x to move from - oo 

 from a, across abk to + oo from a, then the distance ak being 

 denoted by a and bk by p, the anharmonic ratio and distance D 

 have the following values, x being always reckoned from a. 

 x =■ - QO n to - n 

 Pxk= £/« „ + n 



D = 



- co L 



+ n to + (a-/2-0 n ) 



0* 

 Impossible 



CC J 



For the range akb however, ak = a and bk 



x = - oo n to- U 

 Px k = -j8/ai - 0* 

 D = Impossible 



+ n to a to (a + p - 11 ) 

 + U „ 1 „ + cx n 



(a - ft + U to a + to oo 11 



+ GO" „ 1 „ ft* 



+ ao m „ „ -y 



— /3the results are: — 

 (a + P + U ) to + oo n 

 - » n „ -P/a 



Impossible 



These discontinuities are not representable except on an homaloid 



1 The infinitesimal and infinite will be of the same order. 



2 Since it would be the logarithm of a negative number. There is a 

 sense in which it may be called imaginary. 



