Page:Philosophical Transactions of the Royal Society A - Volume 184.djvu/1337

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

is, as before, the intrinsic invariant equation of osculating conics.

So we may, without further investigation, find the similar equation for osculating cubics as we have found it above.

The equation to the osculating curve of each order will aid us in finding that of the next; thus, the general form of a non-singular osculating quartic will be

$\theta _{1}\lambda \left(u_{5}^{4}\lambda ^{3}-\lambda \mu v-\mu ^{3}\right)+\theta _{2}\lambda ^{2}\left(\lambda v+\mu ^{2}\right)+\theta _{3}\left(\lambda v+\mu ^{2}\right)^{2}$
$+\left(\theta _{4}\lambda +\theta _{5}\mu +\theta _{6}v\right)\left\{u_{5}^{4}\lambda ^{2}\left(\mathrm {V} _{8}\lambda +\mathrm {U} _{7}\mu \right)+\left(\mathrm {U} _{7}^{2}\lambda -\mathrm {V} _{8}\mu +\mathrm {U} _{7}v\right)\left(\lambda v+\mu ^{2}\right)\right\}=0$.

The forms of the functions $\theta$ can be found, but they depend upon differential invariants of a higher order than those of which the values have been investigated.