¿jjgfßl№‘8S!t
[146
•ugh E are
f the other
f the lines
angents to
g points of
ts meet in
y definition,
146 J
A MEMOIR ON CURVES OF THE THIRD ORDER.
385
It will be presently seen that the analytical theory leads to the consideration of
a line IJ (not represented in the figure): the line in question is the polar of E
(or F) with respect to the conic which is the first or conic polar of F (or E) with
respect to any syzygetic cubic. The line IJ is a tangent of the Pippian, and more-
the lines EF and IJ are conjugate polars of a curve of the third class
over
syzygetically connected with the Pippian and Quippian, and which is moreover such
that its Hessian is the Pippian.
G, 0 are
the cubic,
of 0, with
The four
bion of the
in question
respect to
s a tangent
the points
ection with
l in three
itact is the
ction with
and more-
conic polar
he first or
esents any
line EFG
conj ugate
jugate pole
tangent to
e tangents
tangent of
harmonics
that the
GFE, and
to FO the
m through
nts to the
efe'. Thus
EO, FO,
ir envelope
Article Nos. 3 to 19.—Analytical investigations, comprising the proof of the
theorems, Article Xo. 2.
3. The analytical theory possesses considerable interest. Take as the equation of
the cubic,
U = x 3 + y 3 + z 3 + 6lxyz = 0 ;
then the equation of the Hessian is
HU = l 2 (x 3 + y 3 + z 3 ) — (1 + 21 3 ) xyz — 0 ;
and the equation of the Pippian in line coordinates (that is, the equation which
expresses that £x + rjy + £z = 0 is a tangent of the curve) is
Ptr=-J(f 3 + *f+£ 3 ) + (- 1 + 4J 3 )^=0.
The equation of the Quippian in line coordinates is
Q U — (1 — 10Z 3 ) (p + y 3 + £ 3 ) — 61- (5 + 4l 3 ) = 0;
and the values of the two invariants of the cubic form are
S=-l+l\
T = 1 - 201 3 - 81 6 ,
values which give identically,
T 2 - 64$ 3 = (1 + 81 3 ) 3 ;
the last-mentioned function being in fact the discriminant.
4. Suppose now that (X, Y, Z) are the coordinates of the point E, and
(X', Y', Z') the coordinates of the point F; then the equations which express that
these points are conjugate poles of the cubic, are
XX'+1 ( YZ' + Y'Z) =0,
YY' +l(ZX' +Z'X) = 0,
ZZ' +l(XY' + X'Y)= 0;
and by eliminating from these equations, first (X', Y', Z'), and then (X, Y, Z), we find
P (.X 3 + Y 3 + Z 3 ) - (1 + 21 3 ) XYZ =0,
P (X' 3 + Y' 3 + Z' 3 ) - (1 + 21 3 ) X' Y'Z' = 0,
which shows that the points E, F are each of them points of the Hessian.
c. ii. 49