341] ON THE SEXTACTIC POINTS OF A PLANE CURVE. 249 
and assuming U=0, 
dm = tPH = — (8»>-6)(3m-T) Яф 6m-U H _ 
(m - l) 2 (m - l) 2 (m - l) 2 
(8.ну = <aay=- да+адая---AL 
(w - l) 2 (on - l) 2 (on - l) 2 
which are for the most part given in my former memoir; the expressions for d->U,. 
d-л V, which are not explicitly given, follow at once from the equations 
(d 1 * + dfiU = 0, (di + 29 x 9 2 + 9 3 ) £7 = 0; 
those for 9id 3 U, d 2 ~U, and dJJ are new, but when the expressions for 9 X 9 3 U and 9 2 2 U 
are known, that for dJJ is at once found from the equation 
(9 X 4 + 69 x 2 9 2 + 49j9 3 + 39 2 2 + 9 4 ) U = 0. 
57. Before going further, I remark that we have identically 
(a, . .\x, y, zf (a,. .ТЦру — vfi, va — Ху, Xfi — pb)- 
ax + hy + gz, hx + by+ fz, gx +fy + cz 
X , p , v 
a fi у 
= (SI,. fi$Xp - ab, pp - fib, vp — ybf, 
(if for shortness p = ax + fiy + yz, b = Xx+py + vz) 
= p 2 (Si, . p, vf 
- 2pb (Si, . .$>, p, v$a, fi, y) 
+ ^ 2 (Si,..][a, fi, 7 ) 2 . 
58. If in this equation we take (a, b, c, f g, h) to be the second differential 
coefficients of U, and write also (a, fi, y)=(d x , 9 ?/ , dfi, the equation becomes 
m (on — 1) UT — (on — l) 2 9 2 = Ф (xd x + yd y -f- zdfi* 
- 2b (хд х + уду + zdfi V 
+ b 2 \J, 
which is a general equation for the transformation of Э 2 (= 9 X 2 ). 
59. If with the two sides of this equation we ojierate on U, we obtain 
on (on — 1) UTU — (on — l) 2 9 2 U — m (on — 1) Ф U 
- 2 (on - 1 )bWU 
+ b"-nU; 
C. V. 
32