PEOFESSOE CAYLEY ON THE SEXTACTIC POINTS OF A PLANE CUEVE. 555 
24. But recollecting that 
Q=(a, 33, G, 4T, fcX&« B f , S,) 2 H 
=(0, 33, C, #, 6, $X< y, d, 2 f, 2</, 2Af), 
and putting 
EO=(3g, ...J (=3Q S ), 
fo=( a, ...x^', ...) (=aOtO, 
we have,jpos£, Nos. 41 to 46, 
— 2(m — l)(m— 2) F% +(3m-7) 2 E^ =(3m-6)(3m-7)HEO 
(m-l)(3m-8)F^ 1 +(3m-7)(3m-8)F^-( m-l) 2 E^ 1 =(3m-6)(3w-7)HFO 
— ii m ~ l)F^i+ (3m— 7)FF - = (3m— 7)OBH, 
and the foregoing equation becomes 
(3m-7)U = -(5m 2 -18m+17)(3m-6) (3m-7)HEO 
-(5m- 9)(m— 2)(3m— 6) (3m-7)HFO 
+ ( m- 2)(25m 2 — 103m— 106)(3m— 7)Oc)H. 
25. But we have 
^Jac.(U, H, Oh)— ( 3m— 6)HEO + (2m— 4)OhH=0, . . . . (J) 
^ Jac.(U, H, Op)-(3m-6)HFO+(3m-6)ObH=0, . . . . (J) 
that is 
3(m-2)HEQ=2(m-2)QdH+&Jac. (U, H, Oh), 
3(m-2)HFO=(3m-8)QhH+& Jac. (U, H, Op), 
and we thus obtain 
n=-(5m 2 -18m+17){2(m-2)QhH+^Jac. (U, H, Oh)} 
— (5m— 9)(m— 2) {(3m-8)OdH+S Jac. (U, H, Op)} 
+(25m 2 — 103m+106)(m— 2)OhH, 
where the coefficient of (m— 2)OdH is 
— (10m 2 — 36m+34) 
— (5m— 9)(3m— 8) 
-j-(25m 2 — 103m+106), 
which is —0. Hence 
H = -(5m 2 -18m+17)^Jac. (U, H, Op) 
— (5m— 9)(m— 2) ^Jac.(U, H, Op). 
26. Substituting this in the equation 
3HH+(m— 2) 2 { — 27H Jac. (U, O, H)+40 Jac. (U, % H)}=0, 
the result contains the factor and, throwing this out, the condition is 
3H { — (5m 2 18m+17) Jac. (U, H, OH)-(5m-9)(m-2) Jac. (U, H, Op)} 
+ (m-2) 2 {27H Jac. (U, H, O)-40 Jac. (U, H, ¥)} = 0, 
5 f 2 
