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XIX. On the Conic of Five-pointic Contact at any point of a Plane Curve. 
By A. Cayley, Esq., F.B.S. 
deceived Marcli 1, — Eead March 24, 1859. 
The tangent is a line passing through two consecutive points of a plane curve, and we 
may in hke 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 Pluckee and Steinee, 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 
T=(#XX, Y, Z)“=0. 
Let {x, y, z) be the coordinates of a given point on the curve, and let U=(#X'^, y, s)"‘ 
be what T 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. 
2. Write for shortness 
DU =(XB,+YB,-f Zd,)U, 
D^U=(Xd.+YB,+ZB,)^U, 
and let n=(35X-l-^Y+cZ = 0 be the equation of a line. It is easy to see that 
D^U-n.DU=0 
will be the equation of a conic having an ordinary (two-pointic) contact with the curve 
at the point (x, y, z). In fact the equation DU = 0 is that of the tangent at the point in 
question, and the equation D^U=0 is that of the penultimate polar (or polar conic) of the 
point, which conic is touched by the tangent ; the assumed equation represents therefore 
a conic having an ordinary (two-pointic) contact with the polar conic, and therefore with 
the curve. It may be added that the two conics intersect besides in a pair of points, and 
that the line joining these, or common chord of the two conics, is the line represented 
by the equation 11=0; and this being so, the constants [a, h, c) of the line 11=0 can 
be so determined as to give rise to a five-pointic contact. 
