556 PROFESSOR CAYLEY ON THE SEXTACTIC POINTS OE A PLANE CURVE, 
or, as this may also be written, 
-(15m 2 -54m+51)HJac.(U, H, Qh)-3(5w-9)(to- 2)H Jac. (U, H, Q v ) 
+ 27(m-2) 2 {H Jac. (U, H, Oh)+ H Jac. (U, H, 0<j)} 
— 40(m— 2) 2 Jac. (U, H, ¥ )=0. 
27. Hence the condition finally is 
(12m 2 -54m+57)H Jac.(U, H, Qfl)+(m-2)(12m-27)H Jac. (U, H, fiy) 
— 40(m — 2) 2 Jac. (U, H, ¥)=0, 
or, as this may also be written, 
-3(m-l)H Jac. (U, H, nH)+(m-2)(12m-27)H Jac. (U, H, H) 
-40 (m-2) 2 Jac. (U, H, ¥)=0, 
viz. the sextactic points are the intersections of the curve m with the curve represented 
by this equation; and observing that U, H, HH and 4" are of the orders m, 3m— 6, 
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 
Q=(a, 33, €, f, <8, ®X3„ 3J’H=-2S . U=0, 
this implies Jac. (U, H, O)=0, and hence if one of the two Jacobians, Jac. (U, H, O^), 
Jac. (U, H, Oh) vanish, the other will also vanish. Now, using the canonical form 
U =x 3 +f+z 3 +6lxyz, 
we have 
0=(<3,. .1 a',...) 
=(yz—l 2 x 2 , zx—l 2 y 2 , xy—l 2 z 2 , l 2 yz—lx 2 , Pzx—ly 2 , l 2 xy—lz 2 X 
x -3 1% -3 l 2 y, -3 1% (1+2 l 3 )x, (1+2 l 3 )y, (1 + 2 l 3 )z), 
the development of which in fact gives the last-mentioned result. But applying this 
formula to the calculation of Jac. (U, H, Qu), then disregarding numerical factors, we 
have 
d,Q ^{yz-Px 2 , . , . Pyz-lx 2 , . , . X-3^ 2 , 0, 0, (1+2Z 3 ), 0, 0) 
= — 3 1 2 ( ; yz-l 2 x 2 ) 
+(1+2 P){l 2 yz-lx>) 
= (-/+^ 2 +2^)=SB a U; 
and in like manner 
and therefore 
Jac. (U, H, Ou)=S Jac. (U, H, U)=0, 
