54 Proceedings of the Royal Irish Academy. 



In order to determine the degree of a curve given by the above 

 equations -sve have only to ascertain the number of points in which 

 an arbitrary line meets it. 



, Let the equation of such a line be Ix + my ^ n% ; and vre have 

 evidently the foUoTving equation to detennine the ratio of \ to /u., or 

 the parameter of each point in which the line meets the curve : — 



(2) lA^ + riiA<i ^ nA^ ^- {IB^^ niB. -f nB^) ^R = 0, 

 or {lAi + riiAo ^ nA^f = {IB^ -f mB^ + nB^fR. 



Now this being an equation of the 2>/i"' degree, the degree of the 

 curve is, in general, 2m. 



We now proceed to investigate the number, and determine the 

 position of, the double points on the curve. 



In order to make clear the spirit of our method we shall first 

 show how the double points on the unicursal cui-ve given by the 

 equations, 



(3} -e ^ A^, 



y = An, 



z =^A,. 



A^, Az, A^, being binary quantics of the m"" degree in X and f/., 

 may be determined. 



Let Z7= be the equation of the curve in x, y, s, which result* 

 from the elimination of the parameter from the above equations ; and 

 let us call L, M, and N, the differential coeflB.cients of U with regard 

 to X, y, and z, respectively. 



"We have then, for any point on the curve Lx + My + Nz = 0, 

 and for the consecutive point Bdx + Mdy + Ndz = 0. 



Hence we easily see that we must have 

 (4) 

 ^ I ^ dx dx\ ,^/' ^ du dy\ ,^ / ^ <fe da\ 



since x, y, and z are homogeneous functions of the »»"" degree 

 in X and //.. 



