658 ME. W. SPOTTISW OODE ON THE SEXTACTIC POINTS OF A PLANE CUEVE. 
2 (n 
j =8. Also since B Z V, B^V, B Z V are 
But since W is of the degree n, ^,=1 + 
linear in x, y, z, it follows that AB,Y=0, AB^Y = 0, AB z V=0; hence, applying the 
formulae (17), (18), 
AflB,Y=£d,V+2(a. • JT. • )« Vi, W)(dJd z V, B„B,V, B*V). 
But since 
9lw' + %)v t + 0u ! = 0 , + (Bu' = H, Bw' -\- Jf r v 1 + Cm' = 0 , 
it follows that 
Similarly, 
so that (20) become 
A«B,V=2B,V+2HB,B,V. 
AwB,y=rB,V+2Hd„B,V, 
qd z Y—rd p Y ■■ 
3H 
: YT< 
3H 
fB z Y — v' B^Y+2^ —2wh)= . . 
rB z V— pB z V= =— («/ B Z Y— w 1 B z V+2wa— 2ug)= . . 
_pB y V-jB # V=^(« 1 B f V-w , d # V+2«^-2m)= . . , 
whence, multiplying by p, q, r respectively, and adding, we have 
0 = 
p 
u x 
B*Y 
+2 
JP 
2 
w' 
a,v 
<? 
r 
if 
B Z Y 
b 
= 
to, B Z V= 
to>, (22) b 
P 
u 
2a— 6u 
=0; 
2 
V 
2 h—6w' 
r 
w 
2 y— to' 
u a 
v h 
w g 
(23) takes the form 
vr—wq=X , wp—ur= Y, uq—vp=Z , 
w.X+w'Y+t/ Z=P 
w'X+^j Y+w' Z=Q 
«/X-f«i' Y-j-w 1 Z=R, ] 
2(aX+AY+yZ)-4P=0; 
( 21 ) 
( 22 ) 
(23) 
(24) 
or finally substituting 2 (ax-\-hy -\-gz)=6u, and forming similar equations in Q and R, 
we have the system 
a(uK—x P ) + h(u Y — yP ) +g(uZ — zP ) = 0 
k(vX— xQ) -j- b(v Y — y Q) -f- g(v Z — zQ ) = 0 
g(wX—x R) +f(wY — yR) + c(wZ — sR) = 0, 
( 25 ) 
