546 PEOEESSOE CAYLEY ON THE SEXTACTIC POINTS OE A PLANE CURVE. 
viz. in +12g we consider as exempt from differentiation (a', b\ d,f',g', H) which depend 
upon H, and in d,Qg we consider as exempt from differentiation ($, 33, C, Jf, <B, fl) 
which depend upon U. We have similarly 
^ 0 =^ 125 +^ 00 , and d+l^+lg+c^Ou ; 
and in like manner 
Jac. (U, H, 0)= Jac. (U, H, O s )+ Jac. (U, H, Qg), 
which explains the signification of the notations Jac. (U, H, Og), Jac. (U, H, Og). 
The condition for a sextactic point is in the first instance obtained in a form involving 
the arbitrary coefficients (A, (a, v) ; viz. we have an equation of the order 5 in (a, [a, v) 
and of the order 12m— 22 in the coordinates (x, y, z ). But writing §-=lx-\-yjy-\-vz, by 
successive transformations we throw out the factors S- 2 , 3-, S-, + thus arriving at a result 
independent of (a, (a, v ) ; viz. this is the before-mentioned equation of the order 12m— 27. 
The difficulty of the investigation consists in obtaining the transformations by means of 
which the equation in its original form is thus divested of these irrelevant factors. 
Article Nos. 1 to 6. — Investigation of the Condition for a Sextactic Point. 
1. Following the course of investigation in my former memoir, I take (X, Y, Z) as 
current coordinates, and I write 
r=(*xx, y, z) m =o 
for the equation of the given curve ; (x, y, z) are the coordinates of a particular point 
on the given curve, viz. the ^sextactic point; and U, =(#$+ y-, z) m , is what T becomes 
when ( x , y, z) are written in place of (X, Y, Z) : we have thus U = 0 as a condition 
satisfied by the coordinates of the point in question. 
2. Writing for shortness 
DU =(XA,+Yd,+Zcg U, 
D 2 U=(Xh ;r +YB 2/ +ZB,) 2 U, 
and taking n=«X+#Y+<?Z = 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 IT are in the former memoir determined so that the 
contact may be a five-pointic one ; the value obtained for II is 
n=f ^DH+ADU, 
where 
A = +(-3Q H +4^). 
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‘ l x-\- : ^d z x-\- 5 xd i x, y+ &c., z+ See. 
