Page:Philosophical Transactions of the Royal Society A - Volume 184.djvu/1336

1212 33. General method of finding the form of the intrinsic invariant equation to a curve of any order.

It will not be necessary to pass through all the steps which I have taken in developing this theory of intrinsic invariant equations.

If we have an equation to a covariant curve, say $$f\left(\pi,\xi\right) = 0$$, and if $$\mathrm{F}\left(\lambda:\mu:v\right) = 0$$, or $$\mathrm{F}\left(\mathrm{X},\mathrm{Y}\right) = 0$$, is the corresponding intrinsic invariant equation where $$\mathrm{X}$$ and $$\mathrm{Y}$$ stand for $$\mu/v$$ and $$\lambda/v$$ respectively, then the relations between $$\pi,\xi$$ and $$\mathrm{X},\mathrm{Y}$$ are essentially of the character of a homographic transformation. Hence, if for points near the origin in the covariant curve $$\pi$$ is written $$\alpha_2h^2 + \alpha_5h^5 + \alpha_7h^7 +\;.\;.\;.\;,$$ and $$h$$ is put for $$\xi$$, for while in the intrinsic invariant equation $$\mathrm{Y}$$ is written $$\mathrm{A}_2h^2 + \mathrm{A}_5h^5 + \mathrm{A}_7h^7 +\;.\;.\;.\;,$$ where $$h$$ is put for $$\mathrm{X}$$, and if $$\mathrm{a}_2$$, $$\mathrm{a}_5$$, $$\mathrm{a}_7$$, &c., are written for the invariantive portions of $$\alpha_2$$, $$\alpha_5$$, $$\alpha_7$$, &c., then $$\mathrm{A}_2$$, $$\mathrm{A}_5$$, $$\mathrm{A}_7$$ differ from the values of $$\mathrm{a}_2$$, $$\mathrm{a}_5$$, $$\mathrm{a}_7$$, &c., by factors of the character which we have considered in the earlier part of this paper.

Thus $$\mathrm{a}_n = \mathrm{D}p^{2n-1}q^{-n-1}\mathrm{A}_n$$, where $$p$$ and $$q$$ stand for the expressions corresponding to those written $$\mu$$ and $$\lambda$$ in (2).

In general $$\mathrm{D} = -u_2^4u_5^3$$, and at the origin

$q = u_2u_5$,$\quad p = 1$.

Therefore,

$\mathrm{a}_n = u_2^4u_5^3\left(u_2u_5\right)^{-n-1}\mathrm{A}_n$, or $\mathrm{A}_n = -u_2^{n-3}u_5^{n-2}\mathrm{a}_n$,

therefore

and, comparing with (33), generally

$\mathrm{A}_n = -u_5^4\mathrm{W}_n$.

Hence the value which, near the origin, is to replace $$\mathrm{Y}$$ is

$-h^2 - u_5^4\left(h^5 + \mathrm{U}_7h^7 + \mathrm{W}_8h^8 + . . .\right).\quad.\quad.\quad.\quad.\quad.\quad$(59).

Taking the general equation of the intrinsic invariant of conics as

$\theta_5\lambda^2 + \theta_4\lambda\mu + \left(\theta_3 + \theta_2\right)\mu^2 + \theta_2\lambda v + \theta_1\mu v + \theta v^2 = 0$,

and substituting for $$\lambda/v$$ and $$\mu/v$$, the values shown above, we find all the invariant coefficients vanish except $$\theta_2$$, and thus