234 
ON THE SEXTACTIC POINTS OF A PLANE CURVE. 
[341 
the development of which in fact gives the last-mentioned result. But applying this 
formula to the calculation of Jac. (U, H, il^), then disregarding numerical factors, we 
have 
d x VL 0 = (yz — Pa?,. ,. Pyz — lx 2 ,. , SI 2 , 0, 0, (1 + 21 3 ), 0, 0) 
= — SP (yz — l 2 x 2 ) 
+ (1 + 21 3 ) (Pyz — laf) 
= (—l + P) (x 2 + 2lyz), = Sd x U; 
and in like manner d y £ljj — Sd y TJ, d z Clu — Sd z U, and therefore 
Jac. (U, H, n d ) = S Jac. (U, H, U) = 0, 
whence also 
Jac. (U, H, il n ) = 0 ; 
and the condition for a sextactic point assumes the more simple form, 
Jac. (U, H, ^) = 0. 
29. Now (former memoir, No. 32) we have 
^ = (2i, 23, e, & x h, d y H, d z Hy 
= (1 + 81 3 ) 2 (y 3 z 3 + z 3 a? + x a y 3 ) 
+ (— 91 6 ) (a? + y 3 + z 3 ) 2 
+ (— 21 — 5P — 20/ 7 ) (x 3 + y 3 + z 3 ) xyz 
+ (- 1 ol 2 - 781 5 + 12Z 8 ) x 2 y 2 z 2 , 
or observing that a? + y 3 + z 3 and xyz, and therefore the last three lines of the expression 
of are functions of U(= x 3 + y 3 + z 3 + Qlxyz) and H (= — l 2 (x 3 + y 3 + z 3 ) + (1 + 2£ 3 ) xyz), 
and consequently give rise to the term =0 in Jac. (U, H, d r ), we may write 
d 7 = (1 + Hi 3 ) 2 (y 3 z 3 + z 3 x 3 + spy 3 ). 
30. We have then, disregarding a constant factor, 
Jac. (U, H, d / ) = Jac. (a? + y 3 + z 3 , xyz, y 3 z 3 + z 3 x 3 + x 3 y 3 ), 
= a? , y 2 , z 2 
yz , zx , xy 
a? (y 3 + z 3 ), y 2 (z 3 + x 3 ), z 2 (a? + y 3 ) 
— x 3 (y 6 — z 3 ) + y 3 (z 3 — x 6 ) + z 3 (a? — y 3 ), 
— (y 3 — z 3 ) (z 3 — x 3 ) (x 3 — y 3 ),