396 Proceedings of the Royal Irish Academy. 



The quartic having four real asymptotes may be described as the 

 locus of the mean point of a tetrahedron Tvhose vertices determine 

 homographic divisions on four given lines. If only two asymptotes 

 are real, the locus is the mean point of a triangle, two of whose 

 vertices determine homographic divisions on two lines parallel to the 

 asymptotes, while the third vertex determines homographic divisions 

 on a conic. Finally, if the curve has no real asymptote, it is the locus 

 of the middle point of a line joining homographic points on two given 

 conies ; or, more generally, it is the locus of a point dividing in a given 

 ratio the line joining corresponding points on a pair of conies homo- 

 graphically divided.' 



To form the equation of an asymptote of p = 2 7, notice that 



the equation of a tangent is 



ci(«?i-0 •-• ^1-^ 



p = 2 7 -1-; ; or, writing x 



"When tQ-e^ 



p = a;ei + 2 — — 



is the equation of the asymptote parallel to ci, the sign 2 including 

 (»- 1) terms. 



Thus, for a conic, the centre is , as the vector to this point is 



$1 — e-i 



on both asymptotes. The equation of the conic referred to its centre 



is easily seen to be 



Cl €>> - t Co Bi — t 

 p - . — h ^^ . , 



Ca-^'i 61 - t Ci—e^ 62 -t 



p = cosh u + sinh u, where e" = — 



^ - «! ^2-^1 61- t 



' Two curves are homographically divided when there is a one-to-one corre- 

 spondence between corresponding points. 



