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X. On the Sextactic Points of a Plane Curve. By A. Cayley, F.B.S. 
Received November 5, — Read December 22 , 1864. 
It is, in my memoir “ On the Conic of Five-pointic Contact at any point of a Plane 
Curve”*', remarked that as in a plane curve there are certain singular points, viz. the 
points of inflexion, where three consecutive points lie in a line, so there are singular 
points where six consecutive points of the curve lie in a conic ; and such a singular 
point is there termed a “sextactic point.” The memoir in question (here cited as 
“ former memoir”) contains the theory of the sextactic points of a cubic curve ; but it is 
only recently that I have succeeded in establishing the theory for a curve of the order m. 
The result arrived at is that the number of sextactic points is =m(12m— 27), the points 
in question being the intersections of the curve m with a curve of the order 12m— 27, 
the equation of which is 
(12m 2 -54m+57)H Jac. (U, H, n s ) 
+ (m— 2)(12m-27)H Jac. (U, H, H g ) 
+40(m-2) 2 Jac. (U, H, ^ )=0, 
where U=0 is the equation of the given curve of the order m, H is the Hessian or 
determinant formed with the second differential coefficients (a, h, c,f g , h) of U, and, 
(91, 33, C, 4f, 1?) being the inverse coefficients (^[=5c— f 2 , &c.), then 
Q=(g, as, e, f, <g, s*) 2 h, 
*=(& 35, €, f, (3, m*rH, B,H, bJI) 2 ; 
and Jac. denotes the Jacobian or functional determinant, viz. " 
Jac. (U, H, V) = 
b„U, dyU, b z U 
b,H, b^H, b s H 
bffl', b/F, b;F 
and Jac. (U. H, O) would of course denote the like derivative of (U, H, Q); the sub- 
scripts (g, u) of O denote restrictions in regard to the differentiation of this function, 
viz. treating Q as a function of U and H, 
Q=(a, 33, c, jr, e, c 'J'> 2 /'> V, 
if (a 1 , V , c',f\ g\ h 1 ) are the second differential coefficients of H, then we have 
b,Q=(b,&, . . X a’,..) (=b,Q g ) 
+ ( a, ..XbX ..) (=b,O g ); 
* Philosophical Transactions, vol. cxlix. (1859) pp. 371 — 400. 
MDCCCLXV. 
5 E 
