146] 
A MEMOIR ON CURVES OF THE THIRD ORDER. 
399 
It is to be remarked that any three conics whatever may be considered as the 
first or conic polars of three properly selected points with respect to a properly selected 
cubic curve. The theorem applies therefore to any three conics whatever, but in this 
case the cubic curve is not given, and the Pippian therefore stands merely for a curve 
of the third class, and the theorem is as follows, viz. the envelope of a line which 
meets any three conics in six points in involution, is a curve of the third class. 
Article No. 23.—Completion of the theory in Liouville, and comparison with analogous 
theorems of Hesse. 
In order to convert the foregoing theorem into its reciprocal, we must replace the 
cubic U = 0 by a curve of the third class, that is we must consider the coordinates 
which enter into the equation as line coordinates; and it of course follows that the 
coordinates which enter into the equation PU = 0 must be considered as point 
coordinates, that is we must consider the Pippian as a curve of the third order: we 
have thus the theorem; The locus of a point such that the tangents drawn from it 
to three given conics (the first or conic poles of any three lines with respect to a 
curve of the third class) form a pencil in involution, is the Pippian considered as a 
curve of the third order. This in fact completes the fundamental theorem in my 
memoirs in Liouville above referred to, and establishes the analogy with Hesse’s results 
in relation to the Hessian; to show this I set out the two series of theorems as 
follows: 
Hesse, in his memoirs On Curves of the Third Order and Curves of the Third 
Class, Crelle, tt. xxvm. xxxvi. and xxxvm. [1844, 1848, 1849], has shown as follows : 
(a) The locus of a point such that its polars with respect to the three conics 
X = 0, Y = 0, Z = 0 (or more generally its polars with respect to all the conics of the 
series \X + pY + vZ = 0) meet in a point, is a curve of the third order F = 0. 
(/,3) Conversely, given a curve of the third order F= 0, there exists a series of 
conics such that the polars with respect to all the conics of any point whatever of 
the curve V= 0, meet in a point. 
(y) The equation of any one of the conics in question is 
dU , dU, dU A 
A. -J H p - J— + V —J- — 0, 
dx dy dz 
that is, the conic is the first or conic polar of a point (\, p, v) with respect to a 
certain curve of the third order U — 0; and this curve is determined by the condition 
that its Hessian is the given curve F=0, that is, we have F = HU. 
(S) The equation V = HU is solved by assuming U — aV +bIIV, for we have then 
II (aV + bHV) = A V + BHV, where A, B are given cubic functions of a, b, and thence 
V — HU = AV + BHV, or A = l, _Z?=0; the latter equation gives what is alone important, 
the ratio a : b\ and it thus appears that there are three distinct series of conics,