518 ON A SPECIAL SEXTIC DEVELOPABLE. [373 
If p = 1, by a single parameter, and the square root of a cubic or quartic function 
of this parameter. 
If p = 2, by a single parameter, and the square root of a quintic or sextic function 
of this parameter. 
If p > 2, by a parameter £, and an algebraical function thereof 7]; where f, rj are 
connected by an equation of the order \ (p + 3) or -|(p 4- 2) according as p is odd or 
even. 
These principles establish a division of plane curves into algebraical classes; all 
plane curves (other than the generating lines) situate on a ruled surface, belong to the 
same algebraical class, and the surface itself belongs to the same class. Hence, if on 
a ruled surface there is either a right line which is not a generating line (this 
cannot be the case for developables) or a conic, or a cubic having a double point, or 
any other plane curve having the maximum number of double points, the surface 
belongs to the class for which p = 0; and in the case of a developable surface the 
equation of the tangent plane may be rationally expressed by means of a single 
parameter; that is, the degree of the planarity is =1, or the surface is planar. This 
leads to the conclusion, that the developable surfaces or torses of the orders 4, 5, 6 
and 7 are all of them planar. 
The author points out that the ‘ special quintic developable ’ of my paper first 
above referred, (viz. that obtained by writing b= 0 in the equation of the sextic 
developable) is in fact the general developable of the fifth order, or quintic torse. 
The foregoing theorem, that for a curve which has the maximum number of 
double points, the coordinates may be expressed rationally by a single parameter, admits 
of a very simple algebraical proof, as is shown in the paper by Clebsch “Ueber 
Curven deren coordinaten rationale Functionen eines Parameters sind,” Crelle, t. lxiv. 
(1864), pp. 43—65. In another paper by the same author, “Ueber die Singularitäten 
algebraischer Curven,” pp. 98—100, it is remarked that if in any plane curve we have 
m the order, n the class, 8 the number of nodes, k of cusps, t of double tangents, 
l of inflexions, then as a deduction from Riemann’s principles, but also at once 
obtainable from Pliicker’s equations, we have 
\ (m — 1) (m — 2) — 8 — k — ^ (n — 1) (n — 2) — t — 
and moreover if from a given curve we derive in any manner another curve, such 
that to each tangent (or point) of the first curve there corresponds a single point (or 
tangent) of the second curve, then in the second curve the expression 
i {m! - 1) (m' - 2) - 8' - = $ (»' - ]) (n' - 2) - r' - o', 
has the same value as in the first curve.