PHILLIPS AND MOORE. — ALGEBRA. OF PROJECTIVE GEOMETRY. 753 



that AiBi = k. The only line through Ai for which distance is of 

 type (2) is the line joining Ai to the intersection of bi and f The 

 intersection of two such lines must be exceptional, and since F is the 

 only exception it follows that along any line through F the distance 

 between two points not on f is zero and conversely if the distance 

 between two non-coincident points is zero, their joining line passes 

 through F. 



We have just shown that if in the relation 



AB = ^ 



A is held fixed, B describes a line b cutting f in a point P on the Hne 

 AF. To determine more exactly this relation between A and b we 

 consider the pairs of points A, B along a line CD not passing through 

 F. On this line A and B satisfy an equation 



B-A=D-C 



where C, D is a particular pair. From the construction for equal 

 vectors it is seen that A, B are corresponding points of a collineation on 

 CD for which P is the only double point. In general to each point A 

 corresponds a line b and for fixed B, A lies on a line. Hence to points 

 A on a line CD correspond lines b through a point. Furthermore, 

 these lines b pass through points B which are projective on CD with A. 

 The correspondence between A and b is therefore a correlation. Since 

 A and B coincide only on f, that line is the locus of points lying on 

 their corresponding lines. To a point P on f corresponds a line through 

 P. The distance from P to any point of this line being finite the dis- 

 tance between ordinary points on the line must be zero. Hence the 

 line passes through F. This is true whether we hold A or B fixed on 

 f Hence F and f are corresponding elements in the correlation. 



The preceding results may be summed up in the statement that if 

 AB and CD are equal, B and D lie on the correspondents of A and 

 C with respect to a correlation in which F and f are corresponding 

 elements and f the coincidence locus. The construction of these cor- 

 relations depends on whether F is or is not on f 



18. Point F on line f. We first consider the case in which the 

 point F is on the line f We have seen that the locus of points at a 

 given distance from a point A is a line through the intersection of 

 FA with f In the present case all such lines pass through F. The 

 correlation that determines equal distance is degenerate with F as 

 singular point. The distance from this point to any point whatever 

 is indeterminate. 



VOL. XLVII. — 48 



