341] 
ON THE SEXTACTIC POINTS OF A PLANE CURVE. 
225 
6. But the equation 
n = f ^DH+ADU, 
which is an identity in regard to (X, Y, Z), gives 
7) P — 2.-2 H 
l ' F H 1 ’ 
d 2 P = Z~d 2 H + Ad 2 U, 
Off- 
8 s P = |ia j if + A9»Ei; 
and substituting these values, the foregoing equation becomes 
2 (0/ + lQdfo + 100^03 +150 1 0 2 2 ) U 
+ (50 4 Ud,H +1003Ud,H+ 100, Ud 3 H) f 4 + A. 20d 2 Ud 3 U= 0; 
X1 
or putting for A its value, = (— 3il# + 4 X F), and multiplying by §# 2 this is 
9# 2 (0! 5 + 1004*0, + 1O0 1 2 03 + 15040, 3 ) U 
+ 15# (d i Ud 1 H+ 2d 3 Ud 2 H+ 20 2 Ud 3 H) 
+ ~ (— 30# + 4 A F). 1O0 2 Ud 3 U=0, 
which is, in its original or unreduced form, the condition for a sextactic point. 
Article Nos. 7 and 8.—Notations and Remarks. 
7. Writing, as in my former memoir, A, B, G for the first differential coefficients 
of U, we have Bv — Gy, GX — Av, Ay, — BX for the values of dx, dy,- dz, and instead 
of the symbol D used in my former memoir, I use indifferently the original symbol 0 1} 
or write instead thereof 0, to denote the resulting value 
04 (= 0) = (Bv - Gy) d x + (GX -Av)d y + (Ay. - BX) d z , 
and I remark here that for any function whatever il, we have 
0il = A, B, G 
X , y, , v 
d x iì, dyi1, 0*0 
= Jac. (U, S-, il), 
where S- = Xx + yy + vz. I write, as in the former memoir, 
= (21, 33, 6, % @, %JX, y, ,) 2 ; 
C. V. 
29