430 
ME. A. CAYLEY’S MEMOIE OX CUETES OE THE THIED OEDEE. 
that is, with respect to any three conics, each of them the first or conic polar of some 
point (X, v) with respect to the cubic. Considering then these three conics, take as 
the equation of the line and let (X, Y, Z) be the coordinates of a point 
of intersection with the first conic, we have 
^x+;;Y+2:z=o, 
X^+2r^Z=0; 
and combining with these a linear equation 
aX4-j(3Y+yZ=0, 
in which (a, (3, 7) are arbitrary quantities, we have 
X: Y: Z = y;j— a^— 7^: — ; 
and hence 
(7^-(3^r+2l(cc^-y^)((3^-cc^)=0, 
an equation in (a, j3, 7) which is in fact the equation in line coordinates of the two points 
of intersection with the first conic. Developing and forming the analogous equations, 
we find 
(-2kl, r, !«, /3, 7>’=0 
( r, -2lC^, e, lln, hi Ia,(3, 7/=0 
( n\ f, -2lln. III. hi. (3, 77 = 0 , 
w'hich are respectively the equations in line coordinates of the three pairs of intersections. 
Now combining these equations with the equation 7=0, we have the equations of 
the pairs of lines joining the points of intersection -with the point (,r=0, y=^). and if 
the six points are in involution, the six lines must also be in involution, or the condition 
for the involution of the six points is 
that is. 
-2hl. l\ HI. =0, 
r, -2llr,. hi. 
f, -In-W 
4ZT?< - In-lO-^-hX+m 
or reducing and throwing out the factor we find 
-z(e+;j^+r)+(-i+ 4 z^)^;;^=o. 
which shows that the line in question is a tangent of the Pippian. 
It is to be remarked that any three conics whatever may be considered as the fii'st or 
conic polars of three properly selected points with respect to a properly selected cubic 
curve. The th^eorem applies therefore to any three conics whatever, but' in this case the 
cubic curve is not given, and the Pippian therefore stands merely for a curve of the third 
class, and the theorem is as follows, viz. the envelope of a line which meets any three 
conics in six points in involution, is a cun^e of the thii*d class. 
