PROFESSOR CAYLEY OFT THE SEXTACTIC POINTS OF A PLANE CURVE. 569 
Appendix, Nos. 54 to 74. 
For the sake of exhibiting in their proper connexion some of the formulae employed 
in the foregoing first transformation of the condition for a sextactic point, I have 
investigated them in the present Appendix, which however is numbered continuously 
with the memoir. 
54. The investigations of my former memoir and the present memoir have reference 
to the operations 
"b g -\-dy 'd !/ -\-dz d z , 
d 2 = d 2 xb x + d 2 yd y + d 2 zb z , 
d 3 =d 3 xd x -\-d 3 yd 2/ -\-d 3 zd l!l , 
&c., 
where if (A, B, C) are the first differential coefficients of a function U = (#]£#, y, z) m , 
and X, (Jj, v are arbitrary constants, then we have 
dx= Bv—Cp, dy=CX—Av, dz=A(i>—B\; 
so that putting 
b=(Bf— (»b, + (Cx-Avfiy + (A^ - 
= A, B, C 
^ , l* , v 
a., a„ * tt 
we have ch = cb The foregoing expressions of (dx, dy, dz) determine of course the 
values of (d 2 x, d 2 y, d 2 z), (d 3 x, d 3 y, d 3 z ), &c., and it is throughout assumed that these 
values are substituted in the symbols d 2 , b 3 , &c., so that d n =d, and d 2 , d 3 , &c. 
denote each of them an operator such as Xb r + Yc^ + ZcL , where (X, Y, Z) are 
functions of the coordinates; such operator, in so far as it is a function of the coor- 
dinates, may therefore be made an operand, and be operated upon by itself or any 
other like operator. 
55. Taking (i a,b,c,f,g,h ) for the second differential coefficients of U, (£1, 33, C, Jf, (3, i?) 
for the inverse coefficients, and FI for the Hessian, I write also 
o> =(&... X*,^ V )\ 
v =(& . . 0Ca„ a„ a.), 
□ =(&... xa« bj, 
S3 =Kx-\-^y-\-vz^ 
Q=(a, ...xa« a„ bjh, =dh, 
T=(a, ...X^H, 3,H, cLH) 2 , 
T =(a, . . ."yjyidg— vb p , vb J —Xd z , Xd^—yid^) 2 , 
5 II 
MDCCCLXV. 
