240 
ON THE SEXTACTIC POINTS OF A PLANE CURVE. 
38. The coefficient of d y H is 
= AdM 4- Bd 2 $ - G (9.21 + dyft), 
which, in virtue of the identity, post, No. 40, 0*21 4- d y S$ 4- 0 Z @ = 0, 
is 
= AdM + 230 z £ + Cd z ® ; 
and in like manner the coefficient of d z H is 
= - (Ad y A 4- Bd y § + Gd y ®\ 
so that the equation is 
(AdM + Bd z % + Gd z ®) d y H ~(Ad y A 4- Bd y $ + Gd z ®) d z H = 0. 
39. But we have 
A a 
4- ¿Qh 
+ ®g 
= H, 
Ah 
4-$b 
+ ®f 
= 0, 
Ag 
+ W 
4- ®c 
= 0, 
or multiplying by x, y, z and adding, 
(m-l)(AA 
+ &B 
4-®G) 
= xH\ 
whence also 
(m — 1) (Ah + $b + ®c + Ad y A + Bdy$ + Cdy®) = ccdyH, 
that is 
(m — 1) (Ad y A 4- Bdy^Q 4- Gd y @) = xd y H ; 
and in like manner 
(m - 1) (Ad z A + Bd z $ + Cd z ®) = xd z H, 
whence the equation in question. The terms in A 2 are thus shown to be equal, and 
it might in a similar manner be shown that the terms in nv are equal ; the other 
terms will then be equal, and we have therefore 
(0. V)iT = Jac. (U, H, <P). 
40. The identity 
dM 4- dy*Q 4- 0 Z @ =0 
assumed in the course of the foregoing proof is easily proved. We have in fact 
d x A 4- 0 ?/ <6 4- d z ® = d x (be ~/ 2 ) 4- d y (fg - ch) + d z (fh - bg) 
- b (d x c - d x g) 4- c (d x b - d y h) +/(- 2d x f+ d y g 4- d z h) 4- g (d y f- d z b) 4- h (- d y c 4- d z f), 
where the coefficients of b, c, f g, h separately vanish : we have of course the system 
d x A 
+ 
4-d z ® =0, 
+ dy% 
4-0,8 =o, 
d x ® 
4- dy% 
4-0# =0.