146] 
A MEMOIR ON CURVES OF THE THIRD ORDER. 
41.1 
satellite point in regard to the Hessian of the tangent at (X, Y, Z). And consider 
the conic 
X (a? + 2lyz) + Y (if + 2Izx) + Z (x 2 + 2lxy), 
which is the first or conic polar of the point (X, Y, Z) in respect of the cubic. The 
polar (in respect to this conic) of the point (P, Q, R) will be 
%x + yy + tQs = 0, 
where 
f = PX + 1 (RY+ QZ), 
y = QY+ l (PZ + RX), 
£=RZ + l (QX + PY); 
or putting for (P, Q, R) their values, 
% = (Y 3 -Z 3 )(X 2 -IYZ), 
v = (Z 3 - X 3 )(Y 2 — IZX), 
f = (Z 3 - P) (Z- — IXY)\ 
and if from these equations and the equation of the Hessian we eliminate (A, Y, Z), 
we shall obtain the equation in line coordinates of the curve which is the envelope 
of the line + yy + = 0. We find, in fact, 
+ v * + = (Y 3 - Z 3 ) (Z 3 - X 3 ) (X 3 - Y 3 ) 
l 3 (X 3 +Y 3 + Zf 
31 (X 3 +Y 3 + Z 3 ) XYZ 
x 
+ 91 2 X 2 Y 2 Z 2 
v + (1 - W)(Y 3 Z 3 + Z 3 X 3 + X 3 Y 3 ), 
=(Y 3 - Z 3 )(Z 3 - X 3 ) (X s - Y 3 ) 
( l 2 (X 3 + Y 3 + Z 3 )XYZ 
| 
x -j + (l-l 3 )X 2 Y 2 Z 2 
[- l (Y 3 Z 3 + Z 3 X 3 + X s Y 3 ); 
and thence recollecting that 
HU = l 2 (X 3 + P + Z 3 ) - (1 4- 21 3 ) XYZ, 
-l(e+t) 3 + £ 3 ) + (- 1 + U 3 ) £r)£= — (Y 3 - Z 3 ) (Z 3 - X 3 ) (X 3 - P) (HU) 2 , 
and the equation of the envelope is 
- I (f + y 3 + £ 3 ) + (- 1 + №) U £ = 0, 
which is therefore the Pippian. We have thus the theorem: 
52—2