KIE. A. CAYLEY’S MEMOIE ON CURVES OE THE THIRD ORDER. 
448 
depending on the points 2, Cl, 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 
(W=0, P=0), (W=0, 0=0), (L=0, P=0), (L=0, 0=0), 
he in the syzygetic cubic T=0. 
The left-hand side of the equation may be written 
-XYZ{a(X^+2rYZ)+^(Y^+2^ZX)-l-7(Z^+2ZXY)}(^^+3/^-f^*+6%z) 
+irj/;z{a(X^+2^YZ)-f/3(Y^+2/ZX)+7(ZH2/XY)}(X^+Y^+Z^-f6ZXYZ); 
and it may be remarked also that we have 
-3XYZ{a(X^+2?YZ)+/3(Y^+2^ZX)+7(Z^+2/XY)} 
equal identically to 
{X(Y^-Z^)(7Y-j3Z)H-Y(Z*-X^)(aZ-7X)+Z(X*-Y*)(j3X-aY)} 
-(aYZ-fi3ZX+7XY)(X^+Y^+Z^-l-6/XYZ). 
Hence if we assume 
X^+Y*+Z^+6ZXYZ=0, 
i 
the equation will take the form 
KU=WL+P0, 
where the constant coefficient K may be expressed under the two different forms 
K=-XYZ{a(X==+2^Z)-l-(3(Y^+2/ZX)+7(Z^+2/XY)} 
=i{X(Y^-Z^)(7Y-/3Z)+Y(Z^-X^)(aZ-7X)+Z(X^-Y*)(i3X-aY)}, 
and W, L, P, 0 have the same values as before. In the present case the point 2 is a 
point of the cubic: the equation W=0 represents the first or conic polar of the point 
in question, and the equation P=0 its second or hne polar, which is also the tangent of 
the cubic. The line L=0 is a line joining the point Cl with the satellite point of the 
tangent at 2, or di'opping altogether the consideration of the point H, is an arbitrary 
line through the sateUite point : the first or conic polar of 2 meets the cubic twice in 
the point 2, and therefore also meets it in four other points ; the conic 0=0 is a conic 
passing through these four points, and completely determined when the particular posi- 
tion of the line through the satelhte point is given. And, as before remarked, 0=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 
;s=«YZ+/ 3ZX4-7XY, X=aX^^-/3W+7Z^ 
3 M 2 
