CONTACT AT ANY POINT OF A PLANE CUEVE. 
389 
37. The five-pointic conic meets the cubic in the point of contact, considered as five 
coincident points, and in a remaining sixth point or point of simple intersection. The 
process by which I originally obtained the equation of the five-pointic conic, led also to 
the equation of the line joining the point of contact with the point of simple intersec- 
tion ; the equation of this line is 
r -f (-2/^-h2Z=)yV) " 
9^3^3^3<j fi-Yj/((l + 8Z%^+(4^+41Z'‘)y^^.r + (— 2/^+2/®)«V) > 
-6Q^2/V{X(3?V-(l-f2Z%^)+Y(3Zy-(l+2Z^>^)-l-Z(3ZV-(l + 2^>^)} 
+ {X(^+2/y0)+Y(3/^-f2fe^)+Z(s^+2%)} = O. 
38. If the conic meet the cubic in six comcident points, that is, if the point of contact 
be a singular point of the kind already spoken of, or, as we may term it, a sextactic point, 
then the last-mentioned line must coincide with the tangent at the point. Represent 
for a moment the equation of the line by 
AX-fBY+CZ=0, 
then this line is to coincide with the line 
{x^+2lyz)lL-\- {f + 2fe^)Y + ( 2 ^-f 2%)Z= 0, 
or we must have 
B(2:^-f2%)-C(/-f2Z^^) = 0, 
C(^^-f2/;y2)-A(2^-h2%)=0, 
A(y' -f 2 — B (.r" + 2 ) = 0, 
which must be equivalent to a single condition. The terms of A, B, C, which contain 
x^-\-1,lyz^ y^-\-2lzx, z^-\-2lxy 
respectively, may, it is clear, be omitted, and omitting also a factor ^xYz\ we may write 
A= 3^^2;((1 + 8/®K+(4^+41Z")a;"y2:4- (— 2Z"-f 2/")3/V) — 2Q(3/V— (1 -f 2/%^), 
B = ‘^>xfz[[\ -f W)if + ( U-\-m)ifzx-\- ( — 2^2+ 2Z®)2V) — 2Q(3^y - (1 J^2iyx), 
and the like value for C. The last of the three equations is 
I — (^^ + 2 Zj/ 2 ) (( 1 -|- + (4/+ All*)y^zx-{- ( — 2 Z" 4- 2 Z®)?/2 V) J 
-2Q{ (/+2Z2.r)(3ZV-(l+2Z®y)-(^^+2Z^.s)(3Zy-(l+2Z^>^)} 
0 , 
where the function on the left hand is 
(l + 8^•)(3:y-^Y+2^(a,•'^-/z)) 
+(4Z-l-41Z*)2Z(^'‘y;s® — xy'^z^) 
+( — 2Z^+2Z'4(4yV — x^yz"^) 
-2Q {-{l+2iyfz-cifz) +U%x^z-fz)], 
3 F 
MDGCCLIX. 
