520 
[374 
374. 
ON THE HIGHER SINGULARITIES OF A PLANE CURVE. 
[From the Quarterly Journal of Pure and Applied Mathematics, vol. vil. (1866), 
pp. 212—223.] 
The theory of the singularities of a plane curve was first established by Pliicker 
in his great work the Theorie der Algebraischen Curven, (1839), where he establishes, 
in regard to the ordinary singularities, a system of six equations; viz. if we have 
to, the order of the curve, 
n, „ 
class, 
£, » 
number of 
double points, 
1C , „ 
cusps, 
T , „ 
double tangents 
tc > >, 
then Plitcker’s six equations are 
n = to (to — 1) — 28 — 3«:, 
l = 3to (to — 2) — 68 — 8/c, 
inflexions, 
t =^m(m - 2) (to 2 — 9) — (to 2 — to — 6) (23 + 3k) + 23 (3 — 1) + 63«: + f k(k — 1), 
to = n(n — 1) — 2t — 3i, 
k = 3n (n — 2) — 6t — St, 
8 = \ n (n — 2) (n 2 — 9) — (w 2 — n — 6) (2t + 30 + 2t (t — 1) + 6-rt + (t — 1), 
equivalent to three equations; thus to and (within proper limits) 8 and k may be 
considered arbitrary, and the first three equations then give n, i, r; and in like 
manner n and (within proper limits) r and i may be considered as arbitrary, and the 
equations then give to, k, 8.