A curve of the third order, or cubic curve, is the locus represented by an equation such as U=(*)(
x
,
y
,
z
)
3
=0; and it appears by my “Third Memoir on Quantics,” that it is proper to consider, in connexion with the curve of the third order U = 0, and its Hessian HU=0 (which is also a curve of the third order), two curves of the third class, viz. the curves represented by the equations PU=0 and QU=0. These equations, I say, represent curves of the third class; in fact, PU and QU are contravariants of U, and therefore, when the variables
x
,
y
,
z
of U are considered as point coordinates, the variables ξ, η, ζ of PU and QU must be considered as line coordinates, and the curves will be curves of the third class. I propose (in analogy with the form of the word Hessian) to call the two curves in question the Pippian and Quippian respectively. A geometrical definition of the Pippian was readily found; the curve is in fact Steiner’s curve R
0
mentioned in the memoir “Allgemeine Eigenschaften der algebraischen Curven,”
Crelle
, t. xlvii. pp. 1-6, in the particular case of a basis-curve of the third order; and I also found that the Pippian might be considered as occurring implicitly in my “Mémoire sur les Courbes du Troisiéme Ordre,”
Liouville
, t. ix. p. 285, and “Nouvelles Remarques sur les Courbes du Troisiéme Ordre,”
Liouville
, t. x. p. 102. As regards the Quippian, I have not succeeded in obtaining a satisfactory geometrical definition; but the search after it led to a variety of theorems, relating chiefly to the first-mentioned curve, and the results of the investigation are contained in the present memoir. Some of these results are due to Mr. Salmon, with whom I was in correspondence on the subject. The character of the results makes it diflicult to develope them in a systematic order; hut the results are given in such connexion one with another as I have been able to present them in. Considering the object of the memoir to be the establishment of a distinct geometrical theory of the Pippian, the leading results will be found summed up in the nine different definitions or modes of generation of the Pippian, given in the concluding number. In the course of the memoir I give some further developments relating to the theory in the memoirs in
Liouville
above referred to, showing its relation to the Pippian, and the analogy with theorems of Hesse in relation to the Hessian. Article No. 1.—
Definitions
,
&c
. 1. It may be convenient to premise as follows:—Considering, in connexion with a curve of the third order or cubic,
a point
, we have— (
a
) The
first or conic polar
of the point. (
b
) The
second or line polar
of the point. The meaning of these terms is well known, and they require no explanation.
Positive Solutions for Third-Order Boundary-Value Problems with the Integral Boundary Conditions and Dependence on the First-Order Derivatives