341] 
ON THE SEXTACTIC POINTS OF A PLANE CURVE. 
247 
53. By forming in a similar manner the coefficients of the other terms, it 
appears that 
Jac. (U, H, V)H-(dîl, B', GJ 
\ , fl , v 
d x H, d y H, djl 
or since the determinant is 
we have the required equation, 
Jac. (U, H, V)H = (32l, ...$A', B’, CJ. 
This completes the series of formulae used in the transformations of the condition 
for the sextactic point. 
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 
d 1 = dxd x + dy 3 y + dzd Z) 
do = d?xd x + dr yd y + d?zd z , 
d 3 = d 3 xd x + dhydy + d 3 zd z , 
&c., 
where if (A, B, G) are the first differential coefficients of a function U = (*][&, y, z) m , 
and A, y, v are arbitrary constants, then we have 
dx = Bv — Cy, dy=GX — Av, dz — Ay — BX ; 
3 = {Bv — Gy) d x + (GX — Av) d y + (Ay — BX) d z 
so that putting 
we have 3i = 3. The foregoing expressions of (dx, dy, dz) determine of course the 
values of (d 2 x, d-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 3 2 , 3 3 , &c., so that di, =3, and 3 2 , 3 3 , &c. denote 
each of them an operator such as Xd x +Yd y Zd z , where (X, F, Z) are functions of 
the coordinates; such operator, in so far as it is a function of the coordinates, may 
therefore be made an operand, and be operated upon by itself or any other like 
operator.