CONTACT AT ANY POINT OF A PLANE CUEVE. 
397 
may be written 
H(Hf-3H?4) + 0W=-(l-l-8/=*)VS, 
which gives the requu’ed decomposition, so that © having this value, the conic will have 
a five-pointic contact. Eeducing by the last equation, and throwing out the factor V, 
we find 
- 9tP(l -j- ©H, -f ©^V= 0 
for the equation of the line joining the point of contact with the point of simple inter- 
section. And if in this equation we write H= — and ©=(l-f 8/")"Q, we 
obtain, finally, for the equation of the line in question. 
9P/PS-6Ay^^QH,4-Q2V=0, 
which is the before-mentioned result. 
5o. It only remains to verify the assumed equation 
+ vs + (1 -f u^){ Qw ~ 3syz^^ . ) = o . 
\ve may write 
— Q = + 6 + 3 l^xyz ^ — 
and then observing that 
W=: 2lx YZ-f-. , 
= 3PxX.^-\ (1 + 2/=’>YZ - . • , 
we find at once 
(1 +8Z^)(QW — 3xyz^'^i)= 
{x' + Ux^yz-^12l^x^yz —xy^z^)X^ 
-(l + 8iy 
H- (2lx^-{- 12^Vy2:— 3^yp- 2%V)YZ 
Next writing 
(3/V— (1-f 2Z%2 )x+ , 
V = (x^-{-2lyz)X-\-.., 
S = o^{(l-h8l^)x* + (4:l-\-iiy)x^yz-\-( — 2l^-\-2l^)y^z^}X+ 
and forming the expression for 
the coefficient of X^ is 
—xyzW,-\-Y^, 
-xyz {3/V-(l+2^®)y^}^ 
which is + 
= (1 + 8iyx‘’ -^Uxhjz -\-12f xyz' —xy \^) ; 
the coefficient of YZ is 
■-2a;y0(3/y-(l + 2Z^>^)(3/V- (lJ^2P)xy) 
+ (y^-{-2lzx)z { (1 -f 8/y^ + (4^+ 41/y^^y + ( — 2/"-f 2V’)xy} 
-\-{z -\-2lxy)y{{l-\-3Py +(4^-l-41/^y2.r-f-( — 2/®+2/®k’y}, 
MDCCCLIX. q „ 
