138 



Proceedings of the Royal Irish Academy. 



Again, we may suppose P and Q' to coincide. P' and Q will then 

 coincide, and we get a pair of tangents A', and their chord A^. There are 

 two sucli chords, joining two pairs of cotangential points ^i-^i'- A^'A-^"- 

 These four chords may l)e called the axes of tlie system. 



To any system of quadrangles (AA'Jj is related another system (AA'),, in 

 which the generators Ag, Aj' take tlie place of \, A/. Tlds may be called the 

 coZ^wmr system, and its axes the collinear axes of the other system. They 

 join the points collinear with A..., A^..., namely, AA', A" A'" and 



a,a;, a;'a^". 



Fig. 3. 



Each of the sides PQ, P'Q', PQ', P'Q of a quadrangle meets a pair of the 

 sides of any quadrangle of the collinear system. Thus the quadrangles 

 intersect in eight points, and the vertices lie in fours in eight planes. 



A quadrangle intersects the collinear axes in eight points, two on each 

 axis. Further, each pair of sides intersects the four tangents at the extremi- 

 ties of a pair of axes other than those which it intersects. Thus, in the 

 annexed figure, if PFQQ' be a quadrangle of which aa'a'a'" are the collinear 



