574 PROFESSOR CAYLEY ON THE SEXTACTIC POINTS OE A PLANE CTJRYE. 
66. Proof of expression for B 3 . 
We have 
V; 
and thence operating on the two sides respectively with B 1? =B, we have 
s>= — Al { 3<I>(.'i3„+y3»4- zd,)+ <M . (xxl.+ij'd, + id ,) } 
+^ T {aav+aa.v} ; 
or since 
B .(a;B*-{-?/B i ,-|-zB..)=B, B^=0, 
this is 
67. Proof of expression for B 3 H. 
Operating with B 3 upon H, we have at once 
B 3 H = _3m-6 1 OBH+ _JL (B .V)H. 
3 m — 1 m— 1 1 m— 1 v ' 
The remainder of the present Appendix is preliminary, or relating to the investiga- 
tion of the expressions for B^U and BiB 3 U, used ante. No. 31. 
68. Proof of equation V 2 BU=OBH-HBO. 
We have identically 
that is 
(a ..oo, ^ ...p„ A) ! -[ca ...jx, ^ a,)]* 
={abc— &c.)(a, . . .^vB,,— ^B*, XB S — vb x , [*b x — ^B^) 2 ; 
a>n-V 2 =Hr; 
and then multiplying by B, and with the result operating on U, we find 
Now 
and thence 
and observing that 
OnBU— V 2 BTJ=HTBU. 
□ u=(2, B„ BJ 2 U 
=(& h, c, 2 /, 2 g, 2 h); 
□ BU = (9[, ...Jjba, B5, Be, 2B/, 2 By, 2BA) ; 
*, / 
9, /> c 
