C. V. 
30 
[341 
>m — 7 ) 2 E^\r } 
m — l) 2 ANq} 
i - 7) HEi2, 
i - 7) HFU, 
,-1)UdH, 
(J) 
(J) 
>} 
»} 
341] ON THE SEXTACTIC POINTS OF A PLANE CURVE. 233 
which is = 0. Hence 
n = — (5m 2 — 18m + 17) A Jac. (U, H, Ujj) 
— (5m — 9) (m — 2) S- Jac. (U, H, U&). 
26. Substituting this in the equation 
SHU + (m - 2) 2 {- 27H Jac. (U, U, H) + 40 Jac. (U, % H)) = 0, 
the result contains the factor A, and, throwing this out, the condition is 
3H {— (5m 2 — 18m + 17) Jac. ( U, H, U n ) — (5m — 9) (m - 2) Jac. ( U, H, ilp)} 
+ (m - 2) 2 {27H Jac. ( U, H, U) - 40 Jac. ( U, H, ¥)} = 0, 
or, as this may also be written, 
— (15m 2 — 54m + 51) H Jac. (U, H, O^) — 3 (5m — 9) (m —2) H Jac. (U, H, f2^) 
4- 27 (m - 2) 2 {H Jac. (U, H, Ujj) + H Jac. (U, H, fi^)} 
- 40 (m-2) 2 Jac. (U, H, 'P ) = 0. 
27. Hence the condition finally is 
(12m 2 — 54m + 57) II Jac. (U, H, Qjj) + (m — 2) (12m — 27) H Jac. ( U, H, ilp) 
— 40 (m — 2) 2 Jac. (U, H, 'F) = 0, 
or, as this may also be written, 
— 3 (m — 1) H Jac. (U, H, H^) + (m — 2) (12m— 27) H Jac. (U, H, 12) 
— 40 (m — 2) 2 Jac. ( U, H, 'P) = 0, 
viz. the sextactic points are the intersections of the curve m with the curve represented 
by this equation; and observing that U, H, HU and 4' are of the orders m, 3m — 6 r 
8m —18 respectively, the order of the curve is as above mentioned = 12m —27. 
Article Nos. 28 to 30.—Application to a Cubic. 
28. I have in my former memoir, No. 30, shown that for a cubic curve 
n (SI, 23, ®, & £3&, dy, d z ) 2 H = — 2S. U = 0; 
this implies Jac. (U, II, U) = 0, and hence if one of the two Jacobians, Jac. (U, H, Uu), 
Jac. (U, H, Ujr) vanish, the other will also vanish. Now, using the canonical form 
U — a? -f- y 3 + z 3 f Clxyz, 
we have 
ft = (SI,. .^a', ...) 
= (yz — lW , zx — l~y : , xy — l‘ l z l , l 2 yz — la; 2 , Pzx — ly~ , l 2 xy — lz 2 ) 
( - 3l 2 x, - SPy, - SPz, (1 + 2P) x, (1 + 2P) y, (1 + 2P)z ),