CONTACT AT ANT POINT OF A PLANE CUEVE. 
391 
Theorem. — The point of simple intersection of the cubic and the five-pointic conic is 
the third point of intersection with the cubic, of the line joining the point of contact 
with the second tangential of this point. 
41, I have not sought to verify this theorem by my formulee. I remark, that com- 
bining it with the before-mentioned theorem, the five-pointic conic is completely deter- 
mined as follows ; viz. — 
Theorem. — The five-pointic conic touches the conic at the point of contact (two con- 
ditions) ; it passes through the two points in which the polar conic is intersected by the 
tangent to the cubic at the tangential of the point of contact (two conditions) ; and it 
passes through the point which is the third point of intersection with the cubic of the 
line joining the point of contact with its second tangential. 
42, The construction for the point of simple intersection leads at once to that for the 
sextactic points ; in fact, consider a point having for its tangential a point of infiexion : 
a point of infiexion is its own tangential, and the second tangential of the first-men- 
tioned point is therefore the point of inflexion; the line joining the point with the 
second tangent is therefore the tangent at the point, and the point of simple intersec- 
tion coincides with the point itself, that is, the point in question is a sextactic point. 
43, 1 represent the equation of the five-pointic conic by 
(«, b, c,f,g,hXX, Y, Z)^=0; 
the value of a is 
-f ^ 3 V’o? — ( 1 + 2 l^)y^ ( -f 2 lyz) 
— Qi{x^-\-Tlyzy‘., 
in which equation 
Q —y^z^ + 2 : V -}- o^y ^ — 3 fx^y'^z ^ ; 
or reducing by the equation x^-\-y^-\-z^-\-Uxyz=9, 
=jV+^®( —x^—&lxyz) — 
that is, 
— Q = -f 6 Ix^yz 3 faf'y’^z ^ — 
and we have 
a— 9(1 + 8P)x*y'^z^ 
+ 3^y V(3ZV+ ( - 1 + U^)x‘^yz-2l{1^2F)fz^) 
-\-{x^-\-Ux^yz-\-?>rxYz^—y^z''){x*-\- Ux^yz-^-^^fy'^z^). 
We have in like manner 
2/= 1 8(1 -j- 8V)lx*y^z^ 
+ 3^'>v{(3?y-(l4-2^^)2:;r)(2^+2%) + (3/V-(l-f2?>3/)(/-f2fe^)} 
-2Q(/+2fe^)(2^+2%); 
the coefficient of 8x^y^z^ in the second line is 
= ( _ 1 + -{-z^} - 4:l{l+2l^)x^yz ; 
3 F 2 
