560 PEOFESSOE CAYLEY ON THE SEXTACTIC POINTS OF A PLANE CUEVE. 
33. Calculation of B 2 UB 3 U. 
This is 
•S 4 
(m— l ) 4 
HBH. 
Article Nos. 34 & 35 . — The Jacobian Formula. 
34. In general, if P, Q, 14, S be functions of the degrees p, q, r, s respectively, we 
have identically 
pF, 
qQ, 
rE, 
sS 
= 0, 
B„Q, 
BJt, 
B X S 
V. 
V* 
V, 
B,Q, 
B*E, 
B*S 
or, what is the same thing, 
pT Jac. (Q, E, S)-#Q Jac. (E, S, P)+rE Jac. (S, P, Q)-sS Jac. (P, Q, E) = 0. 
Hence in particular if P=U, and assuming U=0, we have 
— Jac. (E, S, U)+rE Jac. (S, U, Q)-sS Jac. (U, Q, E)=0. 
If moreover Q=B, and therefore q—\, we have 
— ^ Jac. (E, S, U)+rE Jac. (S, U, B)-sSJac. (U, a, E)=0; 
or, as this may also be written, 
— B Jac. (U, E, S)+rE Jac. (U, 3-, S)-sSJac. (U, 3, E)=0; 
that is 
—3 Jac. (U, E, S)4-rE3S-sSBE=0. 
35. Particular cases are 
(2 m— 4) <P3H— (3m — 6)HBO =3Jac. (U, O , H), ante, No. 12, 
(5m-ll)VH3H-(3m-6)HB(VH)=3 Jac. (U, VJI, H), „ 14, 
{2m— 4)V:BH-(3m-6)HB.V 
(5m— 12) OBH— (3m— 6)HBO 
(8m— 18) TBPI-(3m-6)HB¥ 
(2m- 4) OBH— (3m— 6)HEO 
(3m- 8) OBH— (3m— 6)HFO 
=3 Jac. (U, V 
, H), „ 
55 
=3 Jac. (U, 0 
, H), „ 
18, 
=3 Jac. (U, ¥ 
,H), „ 
19, 
=3 Jac. (U, 
, H), „ 
25, 
=3 Jac. (U, Oh 
,H), „ 
„ 
where it is to be observed that in the third of these formulae I have, in accordance with 
the notation before employed, written B . V to denote the result of the operation B per- 
formed on V as operand. I have also written V : BH to show that the operation V is 
not to be performed on the following BH 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 homogeneity of 
the functions, and the orders of these functions : in the former of the two formulae, the 
