44 Proceedings of the Royal Irish Academy. 



IS'ow this is exactly tlie equation we should find to determine the 

 points in which a line 2 = /\"^^' meets the curve; hence it follows 



that the lines 



%-Xr^ = Q , %- A.--J' = 



meet the eurre in points which correspond. 



Zls'ow, these two lines are obviously connected by a 1 , 1 relation 

 and the locus of their intersections is the conic Q, = %^ which we 

 call the Q, conic. 



11. — The eight R poiats lie on the Q conic. Tor, if a line be 

 drawn through meeting the curve in two corresponding points 

 P and P', the lines QP , QP intersect on the Q, conic ; consequently 

 the Q conic must meet the carve in points which coincide with their 

 corresponding points, or in other words tbe Q conic passes through 

 the R points. 



"We shall show how to construct, geometrically, the Q, conic, and 

 consequently to determine, geometrically, the R points. 



We have already shown how to determine the Q, points ; conse- 

 quently, if we draw any three lines through one of the Q points, and 

 through the other Q, point the lines which correspond to them, we 

 determine by their intersection three points, which, with the Q, 

 points, enables us to determine completely the Q, conic. 



The R points are then found by describing the Q, conic as above 

 indicated. 



12. — To di-aw a tangent to the curve at a given point P. Join 

 P to Q and Q' by lines PQ and PQ' , meeting the curve in two sets 

 of three points, one on each line whose arguments are, say, u and v. 



Then we have, if Q refer to the point P, 



I{6) + V + J' = N, j 



(1) 



from which we obtaia by addition, 



2I{6) + u + v + J+J' = -IN, ... (2) 

 or, sum J + J' = 2N, 



2 1{e) + u + v = 0, 



