412 
A MEMOIR ON CURVES OF THE THIRD ORDER. 
[146 
The envelope of the polar of the satellite point in respect to the Hessian of the 
tangent at any point of the Hessian, such polar being in respect of the conic which 
is the first or conic polar of the point of the Hessian in respect of the cubic, is the 
Pippian. 
Article Nos. 37 to 40.—Investigations and theorems relating to the first or conic polar 
of a point of the cubic. 
37. The investigations next following depend on the identical equations 
{« (X* + 2 IYZ) + /3(Y 2 + 2IZX) + 7 (Z 2 + 2IXY)) 
x {— X YZ (x 3 + y 3 + z 3 ) + (X 3 + Y 3 + Z 3 ) xyz} 
= [X (.x 2 + 2lyz) + Y (y 2 + 2Izx) + Z (z 2 + 2Ixy)} 
x {X ( Y 3 - Z 3 ) (yy — (3z) + Y (Z 3 - X 3 ) (az -yx) + Z (X 8 - Y 3 ) (¡3x - ay)} 
+ {¿c (X 2 + 2IYZ) + y (Y 2 + 2IZX) + z (Z 2 + 2IXY)} 
x {-(aYZ + (3ZX + yX Y) (Xx 2 + Yy 2 + Zz 2 ) + (aX 2 + /3Y 2 + yZJ) (X Yz + Yzx + Zxy)}, 
which is easily verified. 
I represent the equation in question by 
XT = WL + P@ ; 
then considering (x, y, z) as current coordinates, and {X, Y, Z) and (or, /3, y) as the 
coordinates of two given points 2 and O, we shall have U = 0 the equation of the 
cubic, W = 0 the equation of the first or conic polar of 2 with respect to the cubic, 
P = 0 the equation of the second or line polar of 2 with respect to the cubic. The 
equation T = 0 is that of a syzygetic cubic passing through the point 2: the 
coordinates of the satellite point in respect to this syzygetic cubic of its tangent at 
2 are 
X (Y 3 - Z 3 ) : Y(Z 3 - X 3 ) : £ (X 3 - Y 3 ); 
and calling the point in question 2', then L — 0 is the equation of a line through 
the points 2', H. The equation © = 0 is that of a conic, viz. the first or conic polar 
of 2 with respect to a certain syzygetic cubic 
— 2 (a YZ + /3ZX + yXY) (x 3 + y 3 + z 3 ) + (aX 2 + /3 Y 2 + yZ 2 ) xyz — 0, 
depending on the points 2, H, or, what is the same thing, the conic 0 = 0 is a 
properly selected conic passing through the points of intersection of the first or conic 
polars of 2 with respect to any two syzygetic cubics; and lastly, K is a constant 
coefficient. The equation expresses that the points of intersection of 
(If = 0, P = 0), (W= 0, 0 = 0), (L = 0, P = 0), (L = 0, 0-0), 
lie in the syzygetic cubic T = 0.