42 Proceedings of the Royal Irish Academy. 



7. — The equation of our quintic being 



Az" -2Bz + AQ = 0, 



an equation wliicli is obviously satisfied by s = , ^ = , it follows 

 that the tangents at the triple point meet the curve again in three 

 points which lie on a right line, this then is the characteristic of 

 our quintic, that the tangents at meet the curve in three points which 

 lie on a rigid line ; these points, we call the A points, and the line 

 joining them meets the curve in two points which we call the 

 Q points. 



It is easy to see that each of the A points has as a cor- 

 responding point, and consequently the arguments of the A poiats 

 are seen to be -iVg, -iVi, and -N^- If now we call Q and Q! , 

 the two points in which the line joining the A points meets the 

 curve, and J and J' their integrals, as above defined, we must have 



J+J'-N=N, ) 



(1) 



or, J+ J' = 2N. ) - 



The Q points play an important part in the geometry of our quintic. 



8. — Any conic drawn through the Q points and meets the 

 curve in five points whose corresponding points lie on a line. 

 This theorem is readily proved as follows : we have 



2 I{6) + J+J' = iV, 



where 2 refers to the five values of the parameters of the points in 

 in which the conic meets the quintic. Now we proved, in the last 

 Article, that 



J+ J' = 2N. 

 Consequently, 



or, 



which proves the theorem, as - I{e) is the value of I{d) for the 

 points corresponding to those in which the conic meets the curve. 



