666 MR. W. SPOTTISWOODE ON THE SEXTACTIC POINTS OE A PLANE CURVE. 
Hence the whole expression 
=zz 2 {zz, zz YB,Z-ZB,Y B^+zz, zz ZB.X-XB.Z B^Y+zz, zz XB.Y-Y B,X B,Z} 
w' v YB,Z-ZB,Y B,X w' v ZB.X-XB.Z B y Y zy' v XB,Y-YB,,X B,Z 
V 1 w YB a Z-ZB*Y B*X v' w ZB 2 X-XB 2 Z B*Y v' w XB, Y-YB*X B*Z 
. W . 2 H YAZ-ZAY AX . st 2 H ZAX-XAZ AY . sr 2 H XAY-YAX AZ 
-fzz 3 B;X B*Y B,Z u 
B,X B^Y B,Z v* 
B 2 X B 2 Y B 2 Z zy 
AX AY AZ tir 2 H ; 
or in virtue of (33), 
— ZZ 2 {ZZ, ZZ — (^P+ZzP,) 
B^X+zz, zz 
-(jP+yPJ 
B^Y+zz, zz 
— (rP+zyPj) 
3.Z} 
zy' y — (pQ+zzQ,) 
B y X zy' v 
— (ffQ+yQi) 
B,Y zy' y 
— (rQ+zyQ,) 
3,Z 
y' zy — (^zR+zzR,) 
B 2 X y' zy 
— (^R+yR,) 
B 2 Y y' zy 
— (rR+zyR,) 
3,Z 
w 2 H zz Jac.(U, H,Ou) AX . et 2 Hz; Jac.(U,H, Oy) AY . ® 2 Hro Jac. (U, H, 0 0 ) AZ 
+zz 3 B.,X B*Y B,Z zz 
B^X B^Y B y Z v 
B,X B 2 Y B 2 Z zy 
AX AY AZ *t 2 H 
=2zz 2 ot 2 H zzj Pj P+zz 2 Jac. (U, H, Q^zz, zz P +zz 2 Jac. (U, H, Ou) zz, zz P+zz 3 B*X B 2 Y B,Z zz 
zy' Q, Q zy' v Q zy' v Q B^X B^Y B^Z z; 
z;' Ej R y' zy R y' w R B,X B 2 Y B 2 Z w 
AX AY AZ kt 2 H. 
But 
*1 P. P=Z(lP 1 +BQ I +4TPi)-Y((gP 1 +4fQ 1 +CB 1 ) 
zy' Q, Q 
y' R, R 
=zz(a 33 C jf # i?)0 2 r)(P 1 Q 1 R 1 )-^(9[ 33 € f 0 M u v ^XP.Q.R,) 
=«(a 33 c # e Mp 2 rXPABi), 
(a 33 C jr <§ £)(«. y, «)P 1 Q 1 R I )= J ^ T H(P 1 af+Q 1 y+B 1 *)=0, 
(a 33 c jr - (a . .)(? ? ^ »•' <z')x 
+(a..)(p^ ^ ^y)Y 
+(a ..)(!» 2 rXa'jp' rjz 
= Jac. (U, H, N^). 
also 
