232
ON THE SEXTACTIC POINTS OF A PLANE CURVE.
[341
or, as this may also be written,
(3m—7)II=—(5m 2 —18m+17){—2(m—1)( m—2)^ T 4 r 1 +(3m-7) 2 £'i|r}
—(5m—9)(m—2) { (m—l)(3m— 8).F\f r 1 + (3m — 7)(3m — 8)-F'F — ( m —1) 2 # V F 1 ]
+(25m 2 —103m+106)(m-2){ —( m— l)i^ v I / 1 4- (3m —7)#'F j
24. But recollecting that
n = (2i, S3, (5, «, ©, £$0*, 0,, d z f H
= (21, S3, ®, & ©, £$a', 6', c', 2/', y, 2/0,
and putting
#O = (02l, ...$ a', ...) (= 0HO,
Fn=( 21, ...$0a', ...) (=000,
we have, _posi, Nos. 41 to 46,
— 2 (m — 1) (m - 2) + (3m - 7) 2 = (3m - 6) (3m - 7) HEO,
(m - 1) (3m - 8)^ + (3m - 7) (3m - 8) F X V - ( m - l) 2 = (3m - 6) (3m - 7)
-( m - l)i^ + (3m-7)#P- = (3m-7)00#,
and the foregoing equation becomes
(3m — 7) II = — (5m 2 — 18m + 17) (3m — 6) (3m — 7) ##0
- (5m - 9) (m - 2) (3m - 6) (3m - 7) HFCI
+ ( m - 2) (25m 2 - 103m - 106) (3m - 7) O0#
25. But we have
^ Jac. (U, H, O i7 ) - (3m - 6) ##0 + (2m - 4) 09# = 0, (J)
^ Jac. ( if, H, HO - (3m - 6) #i^n + (3m - 6) O0# = 0, (J)
that is
3 (m - 2) ##0 = 2 (m - 2) 00# + ^ Jac. ( U, #, 0 5 ),
3 (m - 2) #i^O = (3m - 8) O0# + ^ Jac. ( U, #, O^),
and we thus obtain
n = - ( 5m 2 - 18m + 17) (2 (m - 2) O0# + St Jac. ( U, #, O^)}
— ( 5m - 9) (m - 2) {(3m - 8) O0# + ^ Jac. ( U, #, O^)}
4- (25m 2 - 103m + 106) (m - 2) O0#,
where the coefficient of (m — 2) O0# is
— (10vn 2 - 36m + 34)
— (5m — 9) (3m — 8)
+ (25m 2 - 103m + 106),
