Page:Philosophical Transactions of the Royal Society A - Volume 184.djvu/1335

If $\left(\lambda _{1}:\mu _{1}:v_{1}\right)$ lies on the tangent at the origin, $\mathrm {U} _{7}\lambda _{1}=0$, and for the tangential of the origin $\lambda _{1}=0$, $\mu _{1}:v_{1}=\mathrm {U} _{7}:\mathrm {V} _{8}$.

Hence, on a non-singular cubic we can have neither $\mathrm {U} _{7}=0$ nor $\mathrm {V} _{8}=0$.

Also, on substitution in the equation to the Hessian, we obtain $u_{5}=0$, 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

$\lambda v+\mu ^{2}=0$,

and to the polar conic of the origin is

$\mathrm {U} _{7}^{2}\lambda ^{2}-\mathrm {V} _{8}\lambda \mu +2\mathrm {U} _{7}\lambda v+\mathrm {U} _{7}\mu ^{2}=0$.

So that

$\mathrm {U} _{7}^{2}\lambda -\mathrm {V} _{8}\mu +\mathrm {U} _{7}v=0$

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

$\mathrm {V} _{8}\lambda +\mathrm {U} _{7}\mu =0$,

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

${\frac {\lambda }{\mathrm {U} _{7}^{2}}}={\frac {\mu }{-\mathrm {U} _{7}\mathrm {V} _{8}}}={\frac {v}{-\mathrm {V} _{8}^{2}}}$,

and, therefore, $\mathrm {V} _{8}\mu -\mathrm {U} _{7}v=0$ 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

${\frac {\lambda }{\mathrm {U} _{7}^{2}}}={\frac {\mu }{-u_{5}^{4}\mathrm {U} _{7}}}={\frac {v}{-\left(\mathrm {U} _{7}^{3}+u_{5}^{8}\right)}}$,

and are independent of $\mathrm {V} _{8}$. This point is the corresidual of the eight consecutive points on each of the several cubics in which $\mathrm {V} _{8}$ 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.