562 PROFESSOR CAYLEY ON THE SEXTACTIC POINTS OE A PLANE CURVE. 
so that the equation is 
P^H+Pl^H+P^JH 
= Pg^H + Q^H + Kgb JH, 
or, as this may be written, 
[{Bd.-Cd,)^-(Ca # -AB,)a]B f H 
+[{BB,-caje-(Aa f -BB # )sri^=o. 
38. The coefficient of c^H is 
=AB.a+BB.®-C &&+*#), 
which, in virtue of the identity, post. No. 40, 
^+^1+^=0, 
is 
=AA$+BBJj+Cd*<g. 
And in like manner the coefficient of <3 .11 
= -(A^+B^+C^), 
so that the equation is 
(A^^+BB,i|+CB^H-(A^a+B^+CB^)B,H=0. 
39. But we have 
9[a+^A+<§^=H, 
%h+%b+®f= 0, 
&g+W+® c = 0 ’ 
or multiplying by x, y, z and adding, 
(m-l)(91A+lB+(aC)=^H ; 
(m-l)(m+l^+#c+A^+B^l+C^)=^H, 
that is 
(m-l)(A^+B^l + C^(g)=^H ; 
and in like manner 
(m- l)(Ad.a+Bd.fc + 05.0)= J&JH, 
whence the equation in question. The terms in X 2 are thus shown to be equal, and it 
might in a similar manner be shown that the terms in p are equal ; the other terms will 
then be equal, and we have therefore 
(d . V)H= Jac. (U, H, $). 
40. The identity 
assumed in the course of the foregoing proof is easily proved. We have in fact 
3,3+3 M+^=Wc-f)+^fg-ch)+^(fh-ig) 
=i(d,c-<>, 9 )+c(d,b--d„l i) 
+/(-23/+3,<,+9^)+ 5 -(3/-3,S)+A(-V+3/), 
