CURVE. 
473 
785] 
The table is arranged according to the value of m; and we have on — 0, n — 1, the 
point; m — 1, n = 0, the line; on — 2, n — 2, the conic; of ra=3, the cubic, there are 
three cases, the class being 6, 4, or 3, according as the curve is without singularities, 
or as it has 1 node, or 1 cusp; and so of m = 4, the quartic, there are nine cases, 
where observe that in two of them the class is = 6,—the reduction of class arising from 
two cusps or else from three nodes. The nine cases may be also grouped together 
into four, according as the number of nodes and cusps (8 + k) is = 0, 1, 2, or 3. 
The cases may be divided into sub-cases, by the consideration of compound singu 
larities ; thus when on = 4, n = 6, = 3, the three nodes may be all distinct, which is the 
general case, or two of them may unite together into the singularity called a tacnode, 
or all three may unite together into a triple point, or else into an oscnode. 
We may further consider the inflexions and double tangents, as well in general as 
in regard to cubic and quartic curves. 
The expression for the number of inflexions 3on (on — 2) for a curve of the order 
on was obtained analytically by Plucker, but the theory was first given in a complete 
form by Hesse in the two papers “Ueber die Elimination, u.s.w.,” and “Ueber die 
Wendepuncte der Curven dritter Ordnung ” (Grelle, t. xxviu., 1844); in the latter of 
these the points of inflexion are obtained as the intersections of the curve u = 0 
with the Hessian, or curve A = 0, where A is the determinant formed with the second 
derived functions of u. We have in the Hessian the first instance of a covariant of 
a ternary form. The whole theory of the inflexions of a cubic curve is discussed 
in a very interesting manner by means of the canonical form of the equation 
oc 3 + y 3 + z s + Qlxyz = 0; and in particular a proof is given of Pliicker’s theorem that the 
nine points of inflexion of a cubic curve lie by threes in twelve lines. 
It may be noticed that the nine inflexions of a cubic curve are three real, six 
imaginary; the three real inflexions lie in a line, as was known to Newton and 
Maclaurin. For an acnodal cubic the six imaginary inflexions disappear, and there 
remain three real inflexions lying in a line. For a crunodal cubic, the six inflexions 
which disappear are two of them real, the other four imaginary, and there remain two 
imaginary inflexions and one real inflexion. For a cuspidal cubic the six imaginary 
inflexions and two of the real inflexions disappear, and there remains one real inflexion. 
A quartic curve has 24 inflexions; it was conjectured by Salmon, and has been 
verified recently by Zeuthen, that at most 8 of these are real. 
The expression \m (on — 2) (on 2 — 9) for the number of double tangents of a curve 
of the order on was obtained by Pliicker only as a consequence of his first, second, 
fourth, and fifth equations. An investigation by means of the curve n = 0, which by 
its intersections with the given curve determines the points of contact of the double 
tangents, is indicated by Cayley, “Recherches sur l’elimination et la the'orie des courbes”, 
('Crelle, t. xxxiv., 1847), [53] : and in part carried out by Hesse in the memoir “ Ueber 
Curven dritter Ordnung ” (Crelle, t. xxxvt., 1848). A better process was indicated by 
Salmon in the “ Note on the double tangents to plane curves,” Phil. Mag. 1858; 
considering the on — 2 points in which any tangent to the curve again meets the 
c. XI. 60