38 



Proceedings of the Royal Irish Academy. 



which pass through 0. These values of 6 are plainly the roots of 

 the equation A = 0, which we shall denote by Wi, %, %. 



The constants Cq, Ci, c^ being the same for all lines, their value 

 will remain unaltered if we consider a line, x = 6y, drawn through 

 0. INow, such a line meets the curve in counted three times, and 

 whose parameters on each branch on which it lies are respectively 

 Wi, %, and W3, and in two corresponding points for which 6 is the 

 same but the radical is equal in value and opposite in sign. The sum, 

 then, of the five integrals I^ reduce to io(^i) + -4(%) + -^o(^3) = -^0 say. 

 We find then 



Co = iVJ, , and by parity of reasoning 



(1) 



Hence any line meets the curve in five points such that 



2/oW = N,, 



(2) 



In precisely the same way we may prove that any conic through 

 the trifle point meets the curve in seven points, and so that 



2/oW = ^0, 



(3) 



As these three equations will, in all cases we shall discuss, always 

 obtain together, we shall write them in the briefer form 



it being always understood that this equation 2 1(6) = JV implies 

 three equations, connecting 2/o(^), 2 /j (6) , and 2/2(6), with 

 iVo, Ni, and JV^. 



