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[146 
•ugh E are 
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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