400 A MEMOIR ON CURVES OF THE THIRD ORDER. [146 
each of them having the above-mentioned relation to the given curve of the third 
order V=0. 
In the memoirs in Liouville above referred to, I have in effect shown that— 
(a') The locus of a point such that the tangents from it to three conics, repre 
sented in line coordinates by the equations X = 0, F= 0, Z = 0 (or more generally with 
respect to any three conics of the series \X +/¿Y + vZ = 0) form a pencil in involution, 
is a curve of the third order V= 0. 
(/3') Conversely, given a curve of the third order V = 0, there exists a series of 
conics such that the tangents from any point whatever of the curve to any three of 
the conics, form a pencil in involution. 
Now, considering the coordinates which enter into the equation of the Pippian as 
point coordinates, and consequently the Pippian as a curve of the third order, I am 
able to add as follows: 
(y') The equation in line coordinates of any one of the conics in question is 
_ dU dU 
X ^ + ^dv + V 
dU 
d£ 
= 0, 
that is, the conic is the first or conic polar of a line (A, g, v) with respect to a 
certain curve of the third class U = 0 ; and this curve is determined by the condition 
that its Pippian is the given curve of the third order V = 0, that is, we have 
V= PU. 
(S') The equation V = PU is solved by assuming U=aPV+bQV, for we have 
then P(aPV+ bQV) = A V+ BHV, where A and B are given cubic functions of a, b; 
and thence V=PU = AV+ BHV, or A = 1, B = 0; the latter equation gives what is 
alone important, the ratio a : b; and it thus appears that there are three distinct 
curves of the third class U=0, and therefore (what indeed is shown in the Memoirs 
in Liouville) three distinct series of conics having the above-mentioned relation to the 
given curve of the third order V — 0. 
It is hardly necessary to remark that the preceding theorems, although precisely 
analogous to those of Hesse, are entirely distinct theorems, that is the two series are 
not connected together by any relation of reciprocity. 
Article Nos. 24 to 28.— Various investigations and theorems. 
24. Reverting to the theorem (No. 18), that the lineo-polar envelope of the line 
EF is the pair of lines OE, OF; the line EF is any tangent of the Pippian, hence 
the theorem includes the following one: