XIX. On the conic of five-pointic contact at any point of a plane curve

Abstract

The tangent is a line passing through two consecutive points of a plane curve, and we may in like manner consider the conic which passes through five consecutive points of a plane curve; and as there are certain singular points, viz. the points of inflexion, where three consecutive points of the curve lie in a line, so there are singular points where six consecutive points of the curve lie in a conic. In the particular case where the given curve is a cubic, the last-mentioned species of singular points have been considered by Plücker and Steiner, and in the same particular case, the theory of the conic of five-pointic contact has recently been established by Mr. Salmon. But the general case, where the curve is of any order whatever, has not, so far as I am aware, been hitherto considered;— the establishment of this theory is the object of the present memoir. I. Investigation of the Equation of the Conic of Five-pointic Contact . 1. I take (X, Y, Z) as current coordinates, and I represent the equation of the given curve by ϒ = (*)(X, Y, Z) m = 0. Let ( x, y, z ) be the coordinates of a given point on the curve, and let U = (*)( x, y, z ) m be what ϒ becomes when ( x, y, z ) are written in the place of (X, Y, Z); we have therefore U = 0 as a condition satisfied by the coordinates of the point in question.