COVARIANTS OF PLANE CURATES. 
1207 
The biquadratic for k may be written 
U, {k^ + U,)^ = 
(k^ 
3 I 
and ¥ + U 7 contains the factor u-^. 
Hence, from the equation (48) for x, x or Z; + Lg contains the factor u-, and 
U 7 — 2 ^Lg — Lg^ or U 7 + ^^ — (^ + Lg)^ contains the factor it-^, so that u-^ is a factor 
of the matrix h. 
The matrix of the polar conic of the tangential of the temporary origin is therefore 
k + Lo 
u- 
U, 
- Lg^ -, 
"T »55 
and the tangent at a sextactic point wall pass through the point of inflexion. 
Evidently there can not exist any sextactic point on the cuspidal cubic. 
§ 5. Invariant Coordinates and Invariantal Equations. 
To make it possible to use the methods of this paper to investigate the properties 
of the higher plane curves, it is desirable to effect some gain in brevity and lucidity 
without any sacrifice in generality. This is the object of the system of invariant 
coordinates, which are introduced in this last section. 
The matrix from which the equation to a covariant curve is developed contains 
u^, Lg, and invariant functions. The order of the curve, in general, depends upon the 
mode in which W 4 and Lg enter the matrix, and its characteristics upon the ratios 
between the invariant functions. These ratios have the properties which entitle them 
to be regarded as the invariantal coordinates of the curve whose equation is developed 
from the matrix. In especial, I shall show that this is the case in the equation to a 
covariant straight line, or line of homographic persistence, and, from the consideration 
of such lines, arrive at a definition of invariant coordinates of a jDoint of homographic 
persistence. After showing that we obtain a dual invariant system of point and line 
coordinates, I, ultimately develop a mode of representing a curve by an invariant 
equation, which will be much shorter than the covariant form of the equation which 
we have as yet considered. The coordinates to the curve will be the characteristic 
invariants of the curve, and none of the homographic peculiarities will be sacrificed by 
the abbreviation. 
30. Introduction of invariant coordinates of a homographically persistent point or 
curve^ and of an intrinsic invariant equation to such a curve. 
The general equation to a covariant line takes the form 
