230 
ON THE SEXTACTIC POINTS OF A PLANE CURVE. 
[341 
so that 
5m — 12, — (3m — 6), 'à-Jac. ( U, il, H) =0, 
1 , 1 , d.ilH 
-5 , 1 , W 
or, what is the same thing, 
(8m - 18) W + 6 0 Jac. (U, il, H) + {i0m- 18) 0 {i1H) = 0, 
we have 
r=ff3n -5fi0V = ft -f 9 *Jac. (U, a, fi)a(ilH). 
19. We have also 
(8m — 18) WdH — (3m — 6) Hd'F — & Jac. ( U, 'F, H) — 0, (J) 
that is 
VdH = j-i-â a Jac. ( Ü, '¥, H) + -U OT ~ 2) Hi№, 
and thence 
9HW + 40^0# = 9H 2 dn - 4oHildH + 40^0/T, 
9 (5m — 9) 
4tm — 9 
Hd {UH) + 
60 {in - 2) 
4m — 9 
+ 4^9 {“27^ J ac . (U, H) + 40 Jac. ( U, V, H)}. 
20. The condition thus becomes 
(15m 2 — 54m + 51) (4m — 9) H Jac. (U, V , H) H 
+ 6 (5m — 9) (m — 2) (4 m — 9) H Jac. {U, VIT, H) 
+ 3 (m — 2) {— 3 (5m — 9) (m — 2) Hd {HH) + 20 (m — 2) 2 T/BT - } 
+ (m — 2) 2 S- {— 27H Jac. ( U, il, H) + 40 Jac. ( U, % H)} = 0, 
which for shortness I represent by 
3JTII + (m — 2) 2 ^ {— 27TT Jac. {U, il, H) + 40 Jac. (t/ T , tf)} = 0, 
so that we have 
II = (5m 2 — 18m +17) (4m — 9) Jac. {U, V , H) H 
+ 2 (5m — 9) (m — 2) (4m — 9) Jac. ( U, V H, H) 
+ (m — 2) {— 3 (5m — 9) (m — 2) 0 (iliT)+ 20 (m — 2) 2 3' V F}. 
21. Write 
^ = {W, 23', <r, S', ©', 5, C)\ 
where (J., B, G) are as before the differential coefficients of U, and (a', b', c, /', g', h') 
being the second differential coefficients of H, (21', 33', (£', g', @', ty) are the inverse 
coefficients, viz., 21' = b'c' — f 2 , &c. We have 
— (m — l) 2 0T r 1 = (3m — 6) (3m — 7)0 {ilH) — (3m — 7) 2 0T f {see post, Nos. 41 to 46),