PEOFESSOE CAYLEY ON THE SEXTACTIC POINTS OF A PLANE CUEVE. 563 
where the coefficients of A, c, f g , A separately vanish : we have of course the system 
d,.(S + c^#+ d*C = 0. 
Article Nos. 41 to 46 . — Proof of identities for the fourth transformation. 
41. Consider the coefficients (a, b, c, f g , h) and the inverse set (9L B, C, Jf, <S, 
and the coefficients (a', V , d,f, g', h 1 ), and the inverse set (91', B', C', 4f', (S', ?!)') ; then 
we have identically 
(a, . .Xff, y, zf(% l', . -X«, • •)-(&'> • -1^ +% • -) 2 
=(a', . .X^r, y, z)X<&, . .X«', • •)— (3, • .X«'®+%+^> • -) 2 > 
where (91', . .fa, . .) and (91, . . \a!, . .) stand for 
(3', as', C', J', (S', i'X«, b,c,2f,2g, 2h) 
and 
(a 3 , C , jr , @ , » Jo!, K c, 2 If, 2^, 2A>) 
respectively. 
42. Taking (a, b, c,f g, A), the second differential coefficients of a function U of the 
order to, and in like manner (a', A', c',f, g', h 1 ), the second differential coefficients of a 
function U' of the order to', we have 
to (to - 1)U . (91', . 0C&„ ch) 2 U' - (to - i) 2 (9T, . -X^,U , 3,U , BJJ ) 2 
i)U' . (a, . oca., cg 2 u — (to' — i ) 2 (9i , . .xa.u', a,u', bjj') 2 ; 
and in particular if U' be the Hessian of U, then to'=3to — 6. 
43. Hence writing 
Q =(3, . . X*« ^) 2 H, ^ =(91, . -X^H, ^H, B.H) 2 , 
Q>=(a', • • X*.» ^) 2 U, ^=(91', . -X^U, 3,U, B S U) 2 , 
we have 
TO(TO-l)U0 1 -(TO-l) 2 ^ 1 =(3TO-6)(3TO-7)Hn-(3TO-7) 2 ^; 
or if U=0, then 
-(to - 1) 2 ^=(3to- 6)(3to- 7)HQ-(3to-7) 2 4'; 
whence also 
— (to— l) 2 Bl r 1 =(3TO— 6)(3 to— 7)(HBO+OhH)— (3 to— 7) 2 Bl r , 
which is the formula, ante No. 21. 
44. Recurring to the original formula, since this is an actual identity, we may 
operate on it with the differential symbol ~d on the three assumptions, — 
1. ( a , b, c,f, g , A), (91, B, C, Jf, (S, H) are alone variable. 
2. (i a /, A', c',/', y', A'), (91', B', (S', JT, (S', $?') are alone variable. 
3. ( x , y, z) are alone variable. 
5 g 2 
