PROFESSOR CAYLEY ON THE SEXTACTIC POINTS OF A PLANE CTJRYE. 553 
or what is the same thing, 
(8m-18)W + 6&Jac. (U, O, H)+(10m-18)b(QH)=0, 
we have 
o — Q 
W=H50-50BH= i ^f- 9 ^Jac. (U, Q, H)— 
19. We have also 
(8m-18)TdH-(3m-6)Hd*-SJac. (U, H)=0, (J) 
that is 
= 4 ^ 9 aJaC '( U ’' 5 '’ H )+“ H ^ 
and thence 
9HW+40^dH 
= 9H 2 BO-45HOBH+40^BH 
= - 9J i^r H5(QH) + 6 -2^- ) H3M' 
+ 4^=9 {-27H Jac.(U, a, H) + 40 Jac. (U, % H)}. 
20. The condition thus becomes 
(15m 2 -54m-f51) (4m-9)H Jac. (U, V , H)H 
+6(5m-9)(m-2)(4m-9)HJac. (U, VH, H) 
+ 3(m-2){-3(5m-9)(m-2)Hb(aH)-|-20(m-2) 2 Hb^} 
-f (m— 2) 2 — 27H Jac. (U, Q, H)+40Jac.(U, % H)} = 0, 
which for shortness I represent by 
3HH-|-(m— 2) 2 —2 TIT Jac. (U, Q, H)+40Jac. (U, % H)}=0, 
so that we have 
11= (5m 2 — 18m+17)(4m— 9)Jac.(U, V , H)H 
+2(5m-9)(m-2)(4m-9) Jac. (U, VH, H) 
+(m— 2){ — 3(5m— 9)(m— 2)d(QH)+20(m— 2) 2 S'^ r }. 
21. Write 
*,=(a', 35', C', 0, WXA, B, C) 2 , 
where (A, B, C) are as before the first differential coefficients of U, and (cl, V, c’,f, cj, hi) 
being the second differential coefficients of H, (£f, 15', C', Jf, 0, If)') are the inverse 
coefficients, viz., Q^b'c'—f' 2 , See. We have 
— (m— l) 2 BT r 1 =(3m— 6)(3?w— 7)b(OH)— (3m— 7) 2 B\P > [see post, Nos. 41 to 46), 
that is 
(3m-6)b(OH)=(3m-7)b^-|^^B^„ 
5 p 
MDCCCLXV. 
