474 
CURVE. 
[785 
curve, he showed how to form the equation of a curve of the order (m — 2), giving 
by its intersection with the tangent the points in question; making the tangent touch 
this curve of the order (m — 2), it will be a double tangent of the original curve. 
See Cayley, “ On the Double Tangents of a Plane Curve ”, {Phil. Trans, t. cxlviii., 
1859), [260], and Dersch {Math. Ann. t. vil, 1874). The solution is still in so far 
incomplete that we have no properties of the curve II = 0, to distinguish one such 
curve from the several other curves which pass through the points of contact of the 
double tangents. 
A quartic curve has 28 double tangents, their points of contact determined as the 
intersections of the curve by a curve II = 0 of the order 14, the equation of which 
in a very elegant form was first obtained by Hesse (1849). Investigations in regard 
to them are given by Plticker in the Theorie der algebraisclien Gurven, and in two 
memoirs by Hesse and Steiner {Crelle, t. xlv., 1855), in respect to the triads of double 
tangents which have their points of contact on a conic, and other like relations. It 
was assumed by Plticker that the number of real double tangents might be 28, 16, 
8, 4, or 0, but Zeuthen has recently found that the last case does not exist. 
The Hessian A has just been spoken of as a covariant of the form u; the 
notion of invariants and covariants belongs rather to the form u than to the curve 
n = 0 represented by means of this form; and the theory may be very briefly referred 
to. A curve u—0 may have some invariantive property, viz. a property independent 
of the particular axes of coordinates used in the representation of the curve by its 
equation; for instance, the curve may have a node, and in order to this, a relation, 
say A = 0, must exist between the coefficients of the equation; supposing the axes of 
coordinates altered, so that the equation becomes u' = 0, and writing A' = 0 for the 
relation between the new coefficients, then the relations A = 0, A' = 0, as two different 
expressions of the same geometrical property, must each of them imply the other; 
this can only be the case when A, A' are functions differing only by a constant factor, 
or say, when A is an invariant of u. If, however, the geometrical property requires 
two or more relations between the coefficients, say A = 0, B = 0, &c., then we must 
have between the new coefficients the like relations, A'= 0, B'= 0, &c., and the two 
systems of equations must each of them imply the other; when this is so, the system 
of equations, A = 0, B — 0, &c., is said to be invariantive, but it does not follow that 
A, B, &c., are of necessity invariants of u. Similarly, if we have a curve (7=0 derived 
from the curve u — 0 in a manner independent of the particular axes of coordinates, 
then from the transformed equation u = 0 deriving in like manner the curve U' = 0, 
the two equations U = 0, U' = 0 must each of them imply the other; and when this 
is so, U will be a covariant of u. The case is less frequent, but it may arise, that 
there are covariant systems U = 0, V=0, &c., and U' = 0, V'= 0, &c., each implying the 
other, but where the functions U, V, &c., are not of necessity covariants of u. 
The theory of the invariants and covariants of a ternary cubic function u has been 
studied in detail, and brought into connexion with the cubic curve u— 0; but the 
theory of the invariants and covariants for the next succeeding case, the ternary quartic 
function, is still very incomplete.