Is, j ' 1 
146] A MEMOIR ON CURVES OF THE THIRD ORDER. 413 
The left-hand side of the equation may be written 
- XYZ {a (X 2 + 2ZFX) + /3 (F 2 + 2IZX) + 7 (X 2 + 2IX Y)) (x 3 + y 3 + z 3 + Qlxyz) 
+ xyz {a (X 2 + 21YZ) + /3 (Y 2 + 2IZX) + y (X 2 + 2IX F)} (X 3 + F 3 + X 3 + 6£X FX) ; 
and it may be remarked also that we have 
- 3XYZ {a (X 2 + 2lYZ) + /3 (F 2 + 21ZX) + y (X 2 + 2£XF)} 
equal identically to 
{X (F 3 - X 3 ) (yY-(3Z) + F (Z 3 - X 3 ) («X - yX) + X (X 3 - F 3 ) (/3X - aF)} 
- (aFX + ¡3ZX + yXF) (X 3 + F 3 + X 3 + 6/XFX). 
Hence if we assume 
X 3 + F 3 + X 3 + QIXYZ = 0, 
the equation will take the form 
K U = WL + P©, 
where the constant coefficient K may be expressed under the two different forms 
K = - XYZ {«(X 2 + 21YZ) + /3 (F 2 + 2£XX) + y (X 2 + 2/XF)} 
= i {X(F 3 - X 3 ) ( 7 F- /3X) + F(X 3 - X 3 ) («X - 7 X) + X(X 3 - F 3 )(^3X - aF)}, 
and IF, X, P, ® have the same values as before. In the present case the point X 
is a point of the cubic: the equation IF = 0 represents the first or conic polar of 
the point in question, and the equation P = 0 its second or line polar, Avhich is also 
the tangent of the cubic. The line L = 0 is a line joining the point Ft with the 
satellite point of the tangent at £, or dropping altogether the consideration of the 
point il, is an arbitrary line through the satellite point: the first or conic polar of 
S meets the cubic twice in the point 2, and therefore also meets it in four other 
points; the conic © = 0 is a conic passing through these four points, and com 
pletely determined when the particular position of the line through the satellite 
point is given. And, as before remarked, © = 0 is a conic passing through the points 
of intersection of the first or conic polars of 2 with respect to any two syzygetic 
cubics. We have thus the theorem : 
The first or conic polar of a point of the cubic touches the cubic at this point, 
and besides meets it in four other points; the four points in question are the points 
in which the first or conic polar of the given point in respect of the cubic is 
intersected by the first or conic polar of the same point in respect to any syzygetic 
cubic whatever. 
38. The analytical result may be thus stated : putting 
K = *YZ + /3ZX + yXY, A = olX 2 + /3 F 2 + 7X 2 ,