238 
ON THE SEXTACTIC POINTS OF A PLANE CURVE. 
[341 
Hence in particular if P = IT, and assuming U= 0, we have 
— qQ Jac. (R, S, U) + rR Jac. (S, U, Q) — sS Jac. (U, Q, R) = 0. 
If moreover Q = and therefore # = 1, we have 
— A Jac. (R, S, TJ) + rR Jac. (8, U, Si) — sS Jac. (U, A, R) = 0 ; 
or, as this may also be written, 
— Sy Jac. (U, R, S) + rR Jac. (U, A, S) — sS Jac. (U, A, R) = 0 ; 
that is 
— A Jac. {U, R, 8) + rRdS— sSd R = 0. 
35. Particular cases are 
(2m — 4) 
QdH — (3m — 6) HdQ 
= A Jac. (U, 
® , H), 
ante, No. 12, 
(5m — 11) V HdH — (3m — 6) Hd (V H) 
= A Jac. ( U, 
V£T, H), 
„ 14, 
(2m — 4) 
V : dH — (3m — 6) Hd . V 
= A Jac. (IT, 
V , H), 
» 
(5m — 12) 
Î23H — (3m — 6) Hdi2 
— ^ Jac. ( U, 
n , H), 
18, 
(8m - 18) 
TJH — (3m — 6) 7J3T 
= Jac. ( U, 
v , H), 
19, 
(2m — 4) 
ildH — (3m - 6) HEQ 
— ^ Jac. (IT,' 
QH > H), 
» 25, 
(3m — 8) 
Î23H - (3m - 6) HFil 
= A Jac. ( IT, 
Flu > H \ 
>y yy 
where it is to be observed that in the third of these formulae I have, in accordance 
with the notation before employed, written 3. V to denote the result of the operation 
3 performed on V as operand. I have also written V : 3H to show that the operation 
V is not to be performed on the following 3H as an operand, but that it remains 
as an unperformed operation. As regards the last two equations, it is to be remarked 
that the demonstration in the last preceding number depends merely on the homo 
geneity of the functions, and the orders of these functions: in the former of the two 
formulae, the differentiation of 12 is performed upon 12 in regard to the coordinates 
(,x, y, z) in so far only as they enter through U, and 12 is therefore to be regarded 
as a function of the order 2m — 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 12 is therefore to be regarded as a function of the order 3m —8. The two 
formulae might also be written 
(2m— 4) 123// — (3m — 6) HdTlj { = A Jac. (IT, 12#, H), 
(3m — 8) 123H - (3m — 6) HdfLi- — S Jac. (U, 12 jj, H); 
and it may be noticed that, adding these together, we obtain the foregoing formula, 
(5m - 12) [IdH - (3m - 6) HdVt = A Jac. ( U, 12, H).