ME. W. SPOTTISWOODE ON THE SEXTACTIC POINTS OF A PLANE CURVE. 667 
Hence the whole expression above written 
= 2w 3 jsr 2 Jac. (U, H, ^u)+ H Jac. (U, H, Q^H+w 3 
^X.., • 
But 
and 
b*x 
B,Y 
B,Z u = - H 2 Jac. (U, H, Q*), 
B,X 
B,Y 
B,Z « 
B,X 
B Z Y 
B z Z to 
AX 
AY 
AZ . 
B*X 
B,Y 
B z Z = - Jac. (U, H, ^)- Jac. (U, H, 0 0 ; 
B,X 
B,Y 
B,Z 
B Z X 
B Z Y 
b 2 z 
Hence, finally, the whole expression 
=v? H^Jac. (U, H, y n )+ (^Ef - ‘(i-7^ 8 ) H Jac ’ ( U - H ’ Q)-®.Jac.(U, H, 0„)} 
(U, H, ¥„)— Jac. (U. H, 0„)) - H Jac. (U, H, Q„)j. 
which is therefore divisible by H u 3 . Consequently H is a factor of all the expressions 
H, . . %! . ., which was to be proved. 
Although not absolutely necessary to our argument, it is perhaps worth while to show, 
as may readily be done, that % is divisible by u. Omitting the terms explicitly multi- 
plied by u in the first three columns, the equation becomes 
WjX— B^P WjY—d^P w,Z— B ir zP 
w'X-B^P w'Y-B^P w'Z-B,zP 
t/X-B^P t/Y-B^P t/Z-B z zP 
p X-AtfP+2HB;X p Y-A?/P+2HB Z Y p Z-AzP+2HB z Z vrJEL 
In this the coefficient of w 2 H, 
~P — (Yz — Zy) 
w'B, P 
+ V(Zx-Xz) 
t/B,P 
H-P(Xy-Y^) 1 MjBJ* 
i/B,P 
w,B z P 
1 w'B^P 1 
+P(M 1 X+wY P +« , Z)-P a (arB # P+yB y P+»B lt P)-P 8 , 
which, writing 
= 0 . ) 
(47) 
= -(w-2)(3HK-}-5P 2 )P. 
K —u u x B*P 
v to' B y P 
w v' B„P 
