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A MEMOIR ON CURVES OF THE THIRD ORDER. 
[146 
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 con 
cluding 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. 
Next, considering, in connexion with the cubic, a line— 
(c) The first or conic polars of each point of the line meet in four points, 
which are the four poles of the line. 
(d) The second or line polars of each point of the line envelope a conic, which 
is the lineo-polar envelope of the line. 
And reciprocally coilsidering, in connexion with a curve of the third class, a line, 
we have: 
(e) The first or conic pole of the line. 
(/) The second or point-pole of the line. 
And considering, in connexion with the curve of the third class, a point— 
(g) The first or conic poles of each line through the point touch four lines, 
which are the four polars of the point. 
(h) The second or point poles of each line through the point generate a conic 
which is the point-pole locus of the point. 
But I shall not have occasion in the present memoir to speak of these reciprocal 
figures, except indeed the first or conic pole of the line. 
The term conjugate poles of a cubic is used to denote two points, such that the 
first or conic polar of either of them, with respect to the cubic, is a pair of lines 
passing through the other of them. Reciprocally, the term conjugate polars of a curve 
of the third class denotes two lines, such that the first or conic pole of either of 
them, with respect to the curve of the third class, is a pair of points lying in the 
other of them.