ON THE SEXTACTIC POINTS OF A PLANE CURVE. 
251 
341] 
and then 
(m - 1) (G'dy - B'd z ) = A [(£« - %)d y - {%z - ®x)-d z ] 
+ [(33« - §y) dy - ($z - %x ) d z ] 
+ v [($« - ©2/) dy - (®z - ©«) d z ] 
= \[x (31, ®^d x , dy, d z ) - 31 (xd x + ydy + zd z )] 
+ P [ x (*£>> 53, % ~§d x , dy, d z ) - £> (xd x + ydy + zd z )\ 
+ r [« (®, % , © ~$d x , dy, d z ) - © (xd x + ydy + zd z )] 
— .«(31, y, v^d x , dy, d z ) - (3t, @]£A, y, v) (xd x + ydy + zd z ) ; 
that is 
and so 
whence 
or finally 
(m — l)(C'dy — B'd z ) = «V —(31, «£), @$\ fi, v)(xd x + yd y + zd z ), 
(m — 1) (A'd z — C'd x ) = y V — (£, 33, % $A, ¡x, v) (xd x 4- yd y 4- zd z ), 
(m — 1) (B'd x — A'dy) = z V - (©, $ , © ¡x, v) (xd x 4- yd y 4- zd z ) ; 
(m — 1) d 2 = (X« + /xy + vz) V — (31, ./x, vf (xd x 4- yd y 4- zd z ), 
= ^ V - d> (xd x + ydy 4- zd 2 ) ; 
d2 = ~^~l^^ dx + ^ + ^ + ^l V - 
63. This leads to the expression for do-U; we have 
0o 2 = 
/■ —1)2 (tôx + ydy + zd z y 
2^ 
- / m _ 1 y 2 $ v (®d x + ydy 4- zd z ) 
+ —V 2 - 
+ (m-l) 2 ’ 
and operating herewith on U, we find 
u= m(m 
(m — 1 ) 2 
-*$£$** u 
32—2