COVARIANTS OP PLANE CURVES. 
1211 
If (Xj : /Xj : v-^ lies on the tangent at the origin, = 0, and for the tangential of 
the origin X^ = 0, /x^ ; = Uy : Vg. 
Hence, on a non-singnlar cubic we can have neither Uy = 0 nor Vg = 0. 
Also, on substitution in the equation to the Hessian, we obtain = 0, or, if the 
tangential is a point of inflexion, the origin must be a sextactic point, as is well 
known. 
The equation to the conic of closest contact at the origin is 
Xv + — 0) 
and to the polar conic of the origin is 
So that 
U^^X^ — YgkfjL “h 2U;^Xi^ + U^/x^ = 0. 
Uy^X — Vg/x -{- \J,~v = 0 
is the equation to the common chord of these two conics, and it is the tangent at the 
tangential of the origin. 
The second tangential of the origin lies on 
VgX + U^/x — 0, 
that is, lies on the common chord of the cubic and the conic of the closest contact. 
The coordinates of the point at which this chord meets the cubic again, or the sixth 
point in which the osculating conic meets the cubic, are given by 
\ fJb V 
TI 2 _ TJ V _ vl ’ 
and, therefore, = 0 is the equation to the line joining this point to the 
tangential of the origin. 
The cordinates of the third tangential of the origin are given by 
X /X V 
1^ = _ - (U/ + «,«) ’ 
and are independent of Vg. This jDoint is the corresidual of the eight consecutive points 
on each of the several cubics in which Vg is arbitrary. This is also a known property, 
but the method allows us, with little difficulty, to prove properties of which other 
analytical proofs are laborious. 
7 P 2 
