188 
ON THE DOUBLE TANGENTS OF A PLANE CURVE. 
[260 
which lies on the second or line polar of the point of contact with respect to the 
Hessian. In my “Memoir on Curves of the Third Order,” Phil. Trans, vol. CXLVII. 
(1857), pp. 415—446, art. No. 37, [146], I gave an identical equation relating to the 
tangential of a cubic, but which is not there exhibited in its proper form; this was 
afterwards effected by Mr Salmon, in the paper “ On Curves of the Third Order,” 
Phil. Trans, vol. cxlviii. (1858), pp. 535—541. The equation, as given by Mr Salmon, 
is in the notation of the present memoir, 
-jq . u+^^.pu-iDH.m + H.r = o, 
an equation which in fact puts in evidence the last-mentioned theorem for the tangential 
of a cubic. 
The idea occurred to me of considering, in the case of the higher plane curves, 
the tangentials 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 complete 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 curve, 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 
following analytical determination of the tangentials of any point of the curve. 
Let 
T=(*£X, 7, Z) n = 0 
be the equation of the given curve, (X, F, Z) being current coordinates; and let 
(x, y, z) be the coordinates of a point on the curve, so that we have 
U = (*$*, y, z) n = 0, 
a condition satisfied by the coordinates of the point in question. 
Then the tangent 
V=(Xd x + Yd y + Zd z )U= 0 
at the point (x, y, z), meets the curve besides in (n — 2) points, which are the 
tangentials of the given point (x, y, z), and which are determined as the intersections 
of the tangent V = 0 with a certain curve, 
n = (t$#, y, z) n ~ 2 = 0.