O'SuLi IVAN — Points on the Curve of Intersection of two Quadrics. 137 



and from any otlier point Q on the curve chords be drawn to meet the joining 

 lines, these chords will meet the curve again in a pair of coiresponding points 

 of the same kind. 



Let PA, PA' be generators Aj, A/ of (A) and (A'), then {v. fig. 1) QP is A.,, 

 QW is A.2'. Hence, by (2) above, tlP' are corresponding points of the same 

 kind as A A'. 



[b) If through each of a pair of corresponding points a chord be drawn to 

 meet a given chord, they will pass through a pair of corresponding points of 

 the same kind. Since yIP, A'P' meet the same chord, they are generators of 

 the same species Aj of the same quadrie (A). Since PA, PA' join P to a pair 

 of corresponding points, and PA is A^, PA' is A/, and A' li' is Aj, .•. UP' are 

 corresponding points, from (2) above. 



Fig. 2. 



The configuration formed by four points in a plane, and their cotangentials, 

 may now be looked at from a new point of view, which is sometimes useful. 



Pairs of corresponding points which are connected by generators Aj, Aj', of 

 the same system of the same pair of conjugate quadrics may be regarded as 

 constituting a system of quadrangles, any pair of corresponding points being 

 vertices of one quadrangle of the system, and one only. We may denote 

 such a system by the symbol (AA')i- 



If in a quadrangle PP'QQ' of the system (AA')i we suppose P and Q to 

 coincide, P' and Q' will also coincide, and the quadrangle degenerates into a 

 pair of tangents, which are generators Aj of (A) and their chord of contact, 

 which is a generator \' of (A'). There are two such chords in each system of 

 quadrangles, which join two paii's of cotangential points A A', A" A'". 



