341] 
ON THE SEXTACTIC POINTS OF A PLANE CURVE. 
223 
2. Writing for shortness 
DU =(Xd X +Yd y + Zd z ) U, 
D*u=(Xd x + Yd y + Zd z yu; 
and taking Tl=aX+bY + cZ = 0 for the equation of an arbitrary line, the equation 
D 2 U-UDU=0 
is that of a conic having an ordinary (two-pointic) contact with the curve at the 
point (x, y, z); and the coefficients of II are in the former memoir determined so 
that the contact may be a five-pointic one; the value obtained for II is 
J1 = %~DH+ADU, 
where 
3. This result was obtained by considering the coordinates of a point of the 
curve as functions of a single arbitrary parameter, and taking 
x + dx + \d?x + \d?x + tj\dtx, y + &c., z + &c. 
for the coordinates of a point consecutive to {x, y, z); for the present purpose we 
must go a step further, and write for the coordinates 
x + dx + ^ d 2 x + £ dl'x + d x x + T ^o d 5 x, 
y + dy + \ dhy +1 d 3 y + ^ d A y + ^ dhy, 
z + dz + | d?z + \d?z + d A z + T ^ T) d 5 z. 
4. Hence if 
0, = dx d x + dy d y + dz d z , 0 2 = d 2 x d x + d 2 y d y + d 2 z d z , &c., 
we have, in addition to the equations 
U= 0, 
dJJ= 0, 
(0^ + 20,) U= 0, 
(0j 3 + 30A + 0 3 ) u = o, 
(0! 4 + 60^02 + 40,03 + 30 2 2 + 0 4 ) U = 0, 
of my former memoir, the new equation 
(0J 5 + 1O0, 3 0 2 + 100,203 + 150,0 2 2 + 50,04 + 1O0 2 0 3 + 0 5 ) U -- 0,