572 PROFESSOR CAYLEY ON THE SEXTACTIC POINTS OE A PLANE CURVE. 
we have 
m(m-l)U(a, . vd x H-Xd,H, Xd y H-{*d x H) 2 -(m-l) 2 (dH) 2 
=m' 2 $H 2 -2mMVII+^ 2 (a, . 3 y H, B Z H) 2 ; 
and if H be the Hessian, then writing m'=3m— 6 and putting also U=0, we find the 
before-mentioned expression for (bH) 2 . 
62. Proof of equation 
5.= -^l(*Uj5,+^)+^V. 
We have 
d 2 =B.d = {(Bv-C^)B x +(Cx-A^+(A^-B^}. 
(*(C3 S -B3.) +MA3- C3.) + »(B3.- A3,)), 
which is 
=X(Od,-B^,)+KA^-CB # )+F(B'd # -A'a f ), 
where 
A'=BA=«(Bi/ — Q*)+A(Cx — Av)-\-g(A(jj — Bx) 
=\(hC-gB)+p(gA-aC)+ V (aB-hA), 
with the like values for B' and C'. Substituting the values 
(m— 1)(A, B, C )=(ax+hy+gz, hx+by+fz, gx+fy+cz), 
we have 
(m-l)A'=x((By-^+Kfy-^)+K€y-fz); 
and similarly 
(m— l)C'=x(i^:— %)+^(3S^— %y)+*($x — %), 
and then 
(w-l)(C'd,~B'B,)= \\_{%x-9iy)b y 
+^[(33^— Ifoy^y—a&z—tfxjb^ 
= x[<a, n, a;, a.)-a(aa.+ya,+*a.)] 
+H>(®, 33, fx^, ^)-»(^.+y^+^.)] 
+»W«, 4f, ^ a.)-®(*&,+yB,+*a.)] 
= <8, ...B, /*, d*)— (& <^X X > v ){^ x -\-y^y+zb z )\ 
that is 
(Bi-lXCra,-»a.)=aV-(a, ©, ex\ p, »)(*M-3«,+sa.), and so 
(«i-i)(A^-aaj= y v-fli, 33, jrxx, & 0(*a.+ya,+*&.), 
(j»-l)(Bfc.-A^)=*V-(«, jf, CXA, p, V ){x-b x +yb s +z\); 
whence 
(m— l)d 2 =(X#+^-f^)V— (gl, . . . JX, v) 2 (#d,+^,+zd z ) 
»•= • -At ^,+^,+^)+Ai v. 
or finally 
