ME. A. CAYLEY ON THE DOUBLE TANGENTS OE A PLANE CUEVE. 
195 
The idea occurred to me of considering, in the case of the higher plane curves, the 
of a given point of the curve, viz. the points in which the tangent again meets 
the curve ; for by expressing that two of these tangentials were coincident, we should 
have the condition that the given point is the point of contact of a double tangent. But 
I was not able to complete the solution. 
Finally, Mr. Salmon discovered the equation of a curve of the order m — 2, which by 
its intersections with the tangent at the given point determines the tangentials, and by 
expressing that the curve in question is touched by the tangent, he was led to a com- 
plete solution of the Double-tangent problem. Mr. Salmon’s result is given in the 
note, “ On the Double Tangents to Plane Curves,” in the Philosophical Magazine for 
October 1858. The discovery just referred to led me to the investigations of the present 
memoir, in which it will be seen that I obtain, for a curve of any order whatever, the 
identical equation corresponding to the before-mentioned equation obtained by Mr. 
Salmon in the case of a cubic ; which identical equation puts in evidence the theorem 
as to the tangentials of the cuiwe, and may thus be considered as containing in itself the 
solution of the Double-tangent problem : the identical equation is besides interesting for 
its own sake, as a part of the theory of ternary quantics. 
1. Mr. Salmon’s solution of the problem of double tangents is based upon the follow- 
ing analytical determination of the tangentials of any point of the curve. 
Let 
T=(*XX, Y, Z)”=0 
be the equation of the given curve, (X, Y, Z) being current coordinates ; and let z) 
be the coordinates of a point on the curve, so that we have 
U=(*X^, y, zf=0, 
a condition satisfied by the coordinates of the point in question. 
Then the tangent 
V=(XB,+Y^,+ ZB,)U=0 
at the point (x, y, z), meets the cuiwe besides in (n — 2) points, which are the tangentials 
of the given point (x, y, z), and which are determined as the intersections of the tangent 
V=0 with a certain curve, 
n=(tx^, y, z)”-*=0. 
2. To express the equation of this curve, let Ui, Ug, . . be the successive emanants of 
U, taken with the facients of emanation y^, zj, viz. 
where it should be noticed that the numerical determination is such, 
{x, y, z) for [x^, y^, zj, then Uj, Uj, . . become respectively equal to U. 
2 D 2 
that putting 
Suppose also 
