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
