548 PBOFESSOB CAYLEY ON THE SEXTACTIC POINTS OF A PLANE CHEVE. 
5. This may be written in the form 
-2[(m-l)(B;+10B?9 a +10B^3+15B 1 ^)+(»i-2)(5BA+10B a B3)]U 
+ P( B'+IOB^ + IOB^+ISB^ +5B 1 B 4 +10B 3 B 3 ) U 
+ 5b 4 P( b?+ 63$,+ 43,3,+ 3b 2 2 )U 
+10b 2 P( b?+ 3 BjB 2 )U 
+10b 3 P( BJU)=0; 
or putting for P its value, =2(m— 2), the equation becomes 
- 2(b5+10b$ 2 + 10d$ 3 +15b 1 b 2 )U 
+ 53^(3*+ 6b?b 2 + 43 x 3 a + 3b 2 )U 
+10b 2 P(b?+ 3b,b 2 )U 
+10b 3 P.b 2 U=0; 
or, as this may also be written, 
2(bs+10b?b 2 +10b- 2 b 3 +15b 1 b 2 )U 
+ 5b 4 P . b 4 U + 10b a P . b 3 U + 10b 3 P . b 2 U = 0. 
6. But the equation 
n = -| gDH + ADU, 
which is an identity in regard to (X, Y, Z), gives 
3 1 P=lH 3 ‘ H ’ 
a 3 P=t h3,H+A3 s U, 
3,P=tHa a H+A3,U; 
and substituting these values, the foregoing equation becomes 
2(b* + l0b 2 b 2 +10b 2 b 3 +15b 1 b 2 )U 
+(5b 4 Ub 1 H+10b 3 Ub 2 H + 10b 2 Ub 3 H)f ^ + A.20b 2 Ub 3 U=0 ; 
or putting for A its value, = g^g(— 3nH+4' v P), and multiplying by fH 2 this is 
9H 2 (b* + 10b?b 2 + 10b?b 3 + lSb^U 
+15H (b 4 Ub^H+2b 3 Ub 2 H + 2b 2 Ub 3 H) 
+ (-3QH + 4>P).10b 2 Ub 3 U=0, 
which is, in its original or unreduced form, the condition for a sextactic point. 
