PEOFESSOE CAYLEY ON THE SEXTACTIC POINTS OF A PLANE CTTEVE. 547 
for the coordinates of a point consecutive to ( x , ?/, z) ; for the present purpose we 
must go a step further, and write for the coordinates 
x dx -f - \d 2 x -f- -d?x -j- ~^d i x -j- ^\od 5 x , 
y+dy +\d 2 y +±d 3 y +-^d 4 y +t h>d% 
z-\-dz -\-\d?z -{-^d 3 z -\--^d i z +xio d s z. 
4. Hence if 
B j = tfcrB* + dyb y + dzb g , B 2 = d 2 xd v + d?yb y -+- d?zb„ &c., 
we have, in addition to the equations 
U=0, 
B.U^O, 
(B?+2B 2 )U=0, 
(B?+3d 1 B 2 +S,)U=0, 
(^i + SBfBa-f-dBjBg-f- 3B 2 -|-B 4 )Uz=0, 
of my former memoir, the new equation 
(B?+10^ 2 +10^?B3 + 15B 1 ^ 2 2 +5B 1 B 4 +10B^3+^ 5 )U=0, 
and in addition to the equations, (P —ax-^-by-\-cz), 
- (m-2)B?U+P.-|B 2 U=0 
- ^[(m-l)^H3(m-2)^ 1 b 2 ]U+P.i(^-l-3B 1 B 2 )U+B 1 P.iB 2 U=0, 
-*[(*»- l)(^t + ^A) + (m- 2)^,+ 3BS)]U 
+P-^+6B^ 2 +4a i a 3 + 3B-)U+B 1 P.i{B;+8B 1 ^)U+iB i P.4drU=0, 
giving in the first instance 
P=2(m— 2), 
B P=a diU 
3 9?U 
> -P _ 1 (at ■ + 69?9 2 )U _ 4 a?U (9? + 39 i9 2 )U 
a ~ 2 d?U 9 9?U 9?U 
and leading ultimately to the before-mentioned value of n, we have the new equation 
— wo [(m— 1)(B? + 10B?B 2 + 10B 2 B 3 +153^5) + (m— 2)(5dA + 10B 2 B 3 )]U 
+ P • T 2 o(^i + 1 OB JB 2 + 10BiB 3 4 - 15BjB 2 +5BA+10BA) U 
+ B t P.* (B}+ 6 B 2 B 2 + 4B 1 B 3 + 3B 2 )U 
+iB 2 P. 1 (B»+ 
+iB s P. | B?U= 0 . 
5 e 2 
