[ 34 ] 



IV. 



SOME PEOPERTIES OF A CERTAIN QUINTIC CTJEYE. Br 

 The Eev. W. E. WESTROPP EOBEETS, B.D., E.T.C.D. 



[Eead January 27, 1902.] 



1. — The Curve, the properties of which I treat of in this Paper, 

 is a special case of the class of quintic curves having a triple point. 

 Such curves possess considerable interest, as many of their properties 

 can be ascertained by the known properties of Abelian integrals 

 and functions, and they thus afford us geometrical interpretations 

 of complex mathematical formulae. ■ 



The equation of a quintic curve having a triple point is readily 

 seen to be of the form 



Az'-2Bz+ C = 0; (1) 



the triple point, which we shall denote by 0, being at the point of 

 intersection of the axes x and y. A, being a binary cubic in x and y, 

 £, a quartic and C a quintic in the same variables, and z a line 

 which passes through the points in which the five lines through 

 whose equation is C = meet the curve. 



"We shall now express the coordinates of a point on the curve 

 in terms of a parameter 6. In order to effect this we shall seek 

 expressions for the coordinates of the two points in which the line 

 X = 6y meets the curve. 



Introducing this value of x into the equation of the ciu've, we 

 find, after dividing by y', 



Jz^ - IBzy ^ Cy"" = 0, (2) 



where A h_ what_^ becomes when we put x = 6 and y = 1 , and, 

 similarly, B and C are what B and C become for the same values 

 of X and y. 



