CONTACT AT ANT POINT OP A PLANE CUEVE. 
387 
where 0 is the first of the three functions which may be chosen to represent the 
octicovariant of the cubic. 
33. We have thus . 4 Q 
iV 9 
and thence 
D^U- IgDU^DU^O 
as the equation of the five-pointic conic : the investigation has been conducted by means 
of the canonical form of the equation of the cubic, but the form of the result shows 
that it applies to the equation of the cubic in any form whatever, 
34. If, however, we continue to represent the cubic by the canonical equation 
the result may be further reduced. We have, putting U = 0,or writing ~ ^Ixy 
H = — (1 + '^l^)xyz ; 
moreover, putting U = 0, the Table No. 70 gives 
or substituting for H the last-mentioned value, and putting for shortness 
Q— 3Z^a^^3^V, 
we have 
0 =(1 + 8 ^ 70 ; 
and with these values of H and Q, the equation of the five-pointic conic is 
D^U+ 
where 
2 (1 + SP)xyz 
DH- 
a 
' 2 7 
( I + 8P)a,^y^z^ 
DU)DU=0, 
DU = 3 { + 2 /y 2 :}X -f ( 3 /® -h 2 lzx)Y + ( 2 :^ 2 lxy)Z } 
D^U=6{(X^+2WZ)^+(Y^+2^ZX)^-f(Z^+2ZXY>}, 
or, as it will be convenient to write it 
= 6 (,:r, y, z, lx, ly, feJX, Y, Z)^ 
DH = (3^V- (1 +2Z%2)X+ (3/y- (1 +2Z^)2^-)Y-f (3^V- (1 + 2Z^)a;y)Z ; 
whence, finally, the equation of the five-pointic conic of the cubic 
X^-l-Y^+Z^+6/XYZ=0 
at the point {x, y, z) is 
%l-\-U^)xYz\x, y, z, lx, ly, IzJX, Y, Z)^ 
-f {{x^-\-^lyz)X.-\-{y'^-\-‘llzx)Y-\-{z‘^-\-2lxy)Z] x 
|3a’VV[(3^V-(l-f2^^%2)X+(3/y-(l-f2/^}2a^)Y-h(3^V-(l-f2Z>7/)z]] 
1 -Q[ix^-^2lyz )X + (/+2fe,2^ )Y+(^^+2% )Z](’ 
a result which I had previously obtained by a special method. 
35. But the expression may be exhibited in a different form by a transformation 
