341] 
221 
341. 
<& 
ON THE SEXTACTIC POINTS OF A PLANE CURVE. 
[From the Philosophical Transactions of the Royal Society of London, vol. clv. (for the 
year 1865), pp. 545—578. 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,” Phil. Trans, vol. cxlix. (1859), pp. 371—400, [261], 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, LLf) 
+ (m — 2) (12m — 27) H Jac. ( U, H, VLf) 
+ 40 (m — 2) 2 Jac. ( U, H, "'P ) = 0, 
where (7=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, b, c, f g, h) of U, 
and, (21, 33, Gt, 8> @, «£>) being the inverse coefficients (21 = be — / 2 , &c.), then 
R = (21, 33, <£, & ©, £]&, d yt d z y H, 
^ = (21, 33, G, 8, ®, £]&#, d y H, d z H)>-