The theory of compound singularities will be referred to farther on.

In regard to the ordinary singularities, we have

m,theorder,
n”class,
δ”number of double points,
κ”    ”   cusps,
τ”    ”   double tangents,
ι”    ”   inflections;

and this being so, Plücker’s “six equations” are

(1)n = m (m − 1) − 2 δ − 3κ,
(2)ι = 3m (m − 2) − 6δ − 8κ,
(3)τ = ½m (m − 2) (m² − 9) − (m² − m − 6) (2δ + 3κ) + 2δ (δ − 1) + 6δκ + 9⁄2κ (κ − 1),
(4)m = n (n − 1) − 2τ − 3ι,
(5)κ = 3n (n − 2) − 6τ − 8ι,
(6)δ = ½n (n − 2) (n² − 9) − (n² − n − 6) (2τ + 3ι) + 2τ (τ − 1) + 6τι + 9⁄2ι (ι − i).

It is easy to derive the further forms—

(7)ι − κ = 3 (n − m),
(8)2 (τ − δ) = (n − m) (n + m − 9),
(9)½m (m + 3) − δ − 2κ = ½n (n + 3) − τ − 2ι,
(10)½ (m − 1) (m − 2) − δ − κ = ½ (n − 1) (n − 2) − τ − ι,
(11, 12)m² − 2δ − 3κ = n² − 2τ − 3ι, = m + n,—

the whole system being equivalent to three equations only; and it may be added that using a to denote the equal quantities 3m + ι and 3n + κ everything may be expressed in terms of m, n, a. We have

κ = a − 3n, ι = a − 3m, 2δ = m² − m + 8n − 3a. 2τ = n² − n + 8m − 3a.