PROFESSOR CAYLEY ON THE SEXTACTIC POINTS OF A PLANE CURVE. 565 
then the three equations are 
-2(m-l)(m-2)F^ 1 =(3m-6)(3m-7)HEO-(3m-7) 2 E'F, 
- (m - 1) 2 E^ = (3m-7)(3m - 8)OBH 
+ (3m- 6)(3m— 7)HFO -2(3m-7)(3m- 8)F^, 
-2(m-l)F^ =2(3m-7)QdH-2(3m-7)F% 
whence, adding, we have 
- (m— l) 2 (EJq + 2F'P 1 ) = - (3m- 7) 2 (E^ + 2F*) 
+(3m-6)(3m-7){OBH+H(EQ+FO)} 
(that is 
- (m- l)*d% = - (3m- 7)W + (3m- 6)(3m- 7)B . OH, 
which is right). 
And by linearly combining the three equations, we deduce 
(3m— 6)(3m— 7)HEO=— 2(m— l)(m— 2) F'F, + (3m-7) 2 E*, 
(3m— 7)OBH = -(m-1) F^+(3m-7) F*, 
(3m- 6)(3m- 7)HFQ= (m- l)(3m- 8)F*~ + (3m- 7)(3m- 8)FF- (m-l) 2 E % , 
which are the formulae, ante, No. 24. 
Article Nos. 47 to 50. — Proof of an identity used in the fourth transformation , viz., 
Jac. (U,VH,H)=- 3 , ^Fi' 1 , 
or say 
Jac. (U, H, VH)= (ST, . .JA, B, CJdA, BB, BC). 
47. We have 
v=( 0 , • -IK t>, 3,. 9.) 
=((& & ^ »), (fi, 33, JflA, (6, f, CB. OP- 9,. 3.) : 
or, attending to the effect of the bar as denoting the exemption of the (91, . .) from dif- 
ferentiation, 
Jac. (U, H, VII) = (& % <£!*., [*, v) Jac. (U, H, B X H) 
+0£b 33, 4fB, (a, v) Jac. (U, H, B y H) 
+ (®, f, CB, h v ) Jac. (U, H, B ; H). 
48. Now 
Jac.(U, H, B x H)=^g Jac. (U, tfB x H+yB y H+zB 2 H, B X H), 
and the last-mentioned Jacobian is 
=B X H Jac. (U, x, B x H)+B y H Jac. (U, y , B x H)-f B 2 H Jac. (U, 2 , B X H) 
+y Jac. (U, B y H, B x H)+z Jac. (U, B 2 H, B X H), 
