374] 
ON THE HIGHER SINGULARITIES OF A PLANE CURVE. 
521 
I have used the ordinary expressions double points, cusps, double tangents, inflexions; 
but (using as I have elsewhere done ineunt as the correlative of tangent) it would be 
more precise and symmetrical to say, double ineunts, stationary ineunts, double tangents, 
and stationary tangents. The double ineunt is called also a node, viz. it is a crunode, 
or an acnode, according as the tangents are real or imaginary; and the stationary 
ineunt, or cusp, considered as (what in the theory of point-coordinates it in fact is) 
a particular case of the double ineunt, is a spinode; to render this notation symmetrical, 
we require certain new terms, say link, as the correlative to node, and flex as the 
correlative to cusp; then the double tangent is a link, viz. it is a colink, or an 
allink according as the ineunts upon it (points of contact) are real or imaginary; and 
the stationary tangent (inflexion) or flex, considered as (what in the theory of line- 
coordinates it in fact is) a particular case of the double tangent, is a relink. The 
ordinary singularities of a plane curve would thus be the node, the cusp, the link, 
and the flex; but I shall retain the above-mentioned more usual expressions. 
Deducible from the six equations, we have 
n — to =$(i — k), 
(n — to) (n + to — 9) = 2 (t — 8), 
which are noticed by Plucker; and also the equation 
(in — 1) (to — 2) — 8 — k = \ (n — 1) (n — 2) — t — i, 
recently noticed by M. Clebsch, in connection with Riemann’s investigations on the 
Abelian Integrals; a curve of the order to may have \(m— 1)(to —2) double points, 
reckoning the cusp as a double point, and so a curve of the class n may have 
\ (n — 1) (n — 2) double tangents, reckoning the inflexion as a double tangent; the two 
sides of this equation exhibit therefore, the right-hand side the deficiency of the actual 
from the possible number of double tangents, and the left-hand side the deficiency of 
the actual from the possible number of double points; and these two numbers are 
equal. We have a division into families based on the value of the expressions in 
question, or say on that of \ (to — 1) (to — 2) — 8 — k ; when this is = 0, that is, when 
the curve has its maximum number of double points (reckoning the cusp as a double 
point), the coordinates x, y are expressible rationally in terms of a parameter 6; when 
the number is = 1, they can be expressed rationally in terms of 6 and of the square 
root of a cubic or a quartic function of 6, &c. &c. It thus appears that as well the 
number 8 + k, as the combinations 28 + 3/e and 68 + 8k which enter into Plticker’s 
equations, plays an important part in the theory of the curve; the bearing of this 
remark will be seen in the sequel. 
Plucker considers also some of the higher singularities; it will be convenient to 
mention two of his results. 
No. 76, p. 216. If two branches of a curve touch each other, or more generally 
have a ^-pointic intersection, the point in question is equivalent to g double points, 
C. V. 66