PROFESSOR CAYLEY ON THE SEXTACTIC POINTS OF A PLANE CURVE. 561 
differentiation of Q is performed upon Q. in regard to the coordinates (x, y, z) in so far 
only as they enter through U, and O is therefore to be regarded as a function of the 
order 2 m — 4 ; in the latter of the two formulae the differentiation is to be performed in 
regard to the coordinates in so far only as they enter through H, and Q is therefore to 
be regarded as a function of the order 3m— 8. The two formulae might also be written 
(2m— 4)I2BH— (3m— 6 )HBQh=S- J ac. (U, Q g , H), 
(3m— 8)ObH— (3m— 6)HbO(j=^ Jac. (U, H) ; 
and it may be noticed that, adding these together, we obtain the foregoing formula, 
(5m-12)QBH-(3m-6)HBQ=3- Jac. (U, Q, H). 
Artie 1 ? Nos. 36 to 40 . — Proof of equation (B .V)H=Jac.(U, H, O), 
36. We have 
used in the second transformation. 
v=(g,.ocx,p > op„3„3,) 
=(i».+Sa,+«8,, ®3,+;f3»+C3J>, p, >). 
Also 
B=(Bv — C i w-)b a .+(C?i — Av)d y -l~(Afjti — Bx)B_ 
= XP -f- ftQ -f- vR, 
if for a moment 
P, Q, B=CB 2 ,— BB*, AB,-CB,, Bd x —Ad r 
Hence 
3.v=(px+Q fl +&).(aa,+©3 9 +®3„ i3, + u3, +J fa = , ga.+.fa.+cajex, p, ,), 
viz. coefficient of X 2 
=P8d Jf +P%d,+PGd„ 
and so for the other terms ; whence also in (B.V)H the coefficients of X 2 , &c. are 
(pgB.+pfcB,+p«3jH, & c . 
37. Again, in Jac. (U, H, <b), where <E>=(£1, 13, C, jf, (0, (*, v) 2 , the coefficients 
of X 2 , &c. are Jac. (U, H, 91), &c. ; and hence the assumed equation 
(B .V)IJ=Jac. (U, H, O), 
in regard to the term in X 2 , is 
(Pa3.+Pfc3 f +M3.)H=Jac. (U, H, 3), 
and we have 
Jac. (U, H, 3)= 
A , B , C 
B,H, B^H, B Z H 
, B, , 
8 
= [B. t H(CB,-BBJ+B y H(AB s -CB,)+B,H(BB x -AB,)]g[ 
=(B,H.P+B,H.Q+B,H.R)a; 
5 G 
MDCCCLXV. 
