554 Prof. Cayley — Sextactic Points of a Plane Curve. [Dec. 22, 



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 establish- 

 ing the theory for a curve of the order m. The result arrived at is that 

 the number of sextactic points is —m{\2m—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 



(12^2-54^2 + 57) H Jac. (U, H, 12^) 

 + (w-2) (\2m-27) H Jac. (U,H,%) 

 + 40 (m-2)2 Jac. (U, H, *)=0, 



where U=0 is the equation of the given curve m, H is the Hessian or 

 determinant formed with the second differential coefficients (a, 5, c,/, g, h) 

 of U, and, (^,23, ^, being the inverse coefficients {^=bc—f, &c.), 

 then 



and Jac. denotes the Jacobian or functional determinant, viz. 



Jac. (U,H,^) = 



"^x ^, ^2 ^ 



and Jac. (U, H, O) would of course denote the like derivative of (U, H, ii) ; 

 the subscripts (H, tj) of Q, denote restrictions in regard to the differentia- 

 tion of this function, viz. treating O as a function of U and H, 



^, ^Ja', b', c\ 2/, 2g\ 2h' ) ; 



if (a', h'y c',f, g'y h') are the second differential coefficients of H, then we 

 have 



viz. in 'bx we consider as exempt from differentiation (a', b', c',f', g'y h') 

 which depend upon H, and in "bx we consider as exempt from differen- 

 tiation C^, S, 5^) which depend upon U. "We have similarly 

 by 12,=dy Og +dy O^, and "bz 12 Og +9^ ; and in like manner 



Jac. (U, H, Q) = Jac. (U, H, %) + Jac. (U, H, a^), 

 which explains the signification of the notations Jac. (U, H, Dg), Jac. 

 (U,H,%). 



The condition for a sextactic point is in the first instance obtained in a 

 form involving the arbitrary coefficients (X, n, v) ; viz. we have an equation 

 of the order 5 in (\, fx, v) and of the order 12m— 22 in the coordinates 

 (x, y, sr). But writing ^=\x-^ /jiy-^pz, by successive transformations we 



