PROFESS OE CAYLEY ON THE SEXTACTIC POINTS OF A PLANE CURVE. 571 
known, that for d 4 U is at once found from the equation 
(^+6B$ 1 +4B 1 3,+ 3a;+BJU=0. 
57. Before going further, I remark that we have identically 
(a, . .)>, y, z)\a , . .\yy— v(3, m—Xy, X^-yuf 
— ax+hy+gz , hx+by+fz, gx+fy+cz 2 
X , fo , v 
“ P 7 
= ( 91 , • -Xxp — vp — y§) 2 , 
(if for shortness j9=ax-^~Py-\-yz, §='kx-\-yjy-\-vz) 
= 
—2p&(8, . .J\, yj, *}>, P, V) 
+V(%,..j a ,p,yy. 
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, /3, y) = (d x , <3 y , cL), the equation becomes 
m{m— l)Ur — (m— 1) 2 B 2 = $(xb x -{-yby-\-zb z y‘ 
—2 
+S 2 D, 
which is a general equation for the transformation of B 2 (=df). 
59. If with the two sides of this equation we operate on U, we obtain 
m(m— l)UrU — (m— l) 2 d 2 U = m(m-l)ffiU 
— 2(m— 1)WU 
+^ 2 DU; 
and substituting the values 
FU=2d>, VU=^ t H, □U=3H, 
we find the before-mentioned expression of h 2 U. 
60. Operating with the two sides of the same equation on a function H of the order 
m f , we find 
m(m- l)UrH - (m- l) 2 b 2 H= 
-2(m'-l)WH 
+ S 2 DH; 
and in particular if H is the Hessian, then writing m'=3m— 6, and putting U=0, we 
find the before-mentioned expression for d 2 H. 
61. But we may also from the general identical equation deduce the expression for 
(dH) 2 . In fact taking H a function of the degree m' and writing 
(*, 1 3, y)=(3 x H, 3,H, bJE), 
5 h 2 
