398 
A MEMOIR ON CURVES OF THE THIRD ORDER. 
[14(5 
it is true with respect to any three conics whatever of the form 
, dU dJJ 
A —j—b y j h v 
dx dy 
dU 
dz 
= 0, 
that is, with respect to any three conics, each of them the first or conic polar of 
some point (A, y, v) with respect to the cubic. Considering then these three conics, 
take + yy 4- %z = 0 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 + yY+ ÇZ = 0, 
X 2 +2IYZ =0; 
and combining with these a linear equation 
aX + /3Y+ 7 Z = 0, 
in which (a, /3, 7) are arbitrary quantities, we have 
X : Y : Z=yr)~№ : a£-y% : /3£ - ar;; 
and hence 
(m ~ fit) 2 + 21 « ~ 7£) - a v) = 0, 
an equation in (a, /3, 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 
(-2% £ 2 > V 2 , -tf-lp, l&i , m 5«, /3, y) 2 = 0, 
( £ 2 > ~ I 2 > > - £? - h 2 , hK /3, 7) 2 = 0, 
( v 2 , £ 2 , -2l£v> %£ > h? > -£y-l? 2 ][ a > fi> y) 2 = 0, 
which 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 (x = 0, y= 0), 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 
-2% ? , m 
£ 2 , - 2IÇq, lr)Ç, 
V 2 , I 2 , - 
= 0, 
that is, 
(- h -1£ 2 ) + hV + + 2l 2 ?rf¥ + 2pfyf 8 + ? (- g v - l?) = 0 ; 
or, reducing and throwing out the factor £ 3 , we find 
- I (f 8 + + £ 3 ) + (-1 + 4>l 3 ) 0, 
which shows that the line in question is a tangent of the Pippian.