ME. W. SPOTTIS W OODE ON THE SEXTACTIC POINTS OE A PLANE CURVE. 669 
out those factors, the three expressions %, JB, ^ are of the form 
Am 2 +B,w +C 1 =0,| 
Kv 2 +B 2 v + C 2 =0,1 (49) 
Aw 2 -fiB 3 w+C 3 =0,j 
in which the coefficients of u 2 , v 2 , w 2 are the same, viz. the expressions given in (46). 
From these equations it follows that 
BjW + Cj B 2 W + C 2 BgW -f Cg 
~77 7 77 • { } 
But as u, v, w do not in general vanish simultaneously, these relations can hold good 
only in virtue of B, being divisible by u x and C, by u 2 ; B 2 by v, and C 2 by v 2 ; B 3 by w and 
C 3 by w 2 . Whence, finally, % is divisible by H u 3 , JB by Hw 3 , ^ by Hw 3 ; and y he 
degree of the equation is reduced to 
(18»-36)-3(»-2)-3(»-l)=12»-27. 
Also, since the ratios (B^+Cj) : u 2 , (B 2 -y+C 2 ) : v 2 , (B 3 w+C 3 ) : w 2 are in virtue of (50) 
equal (say =B), it follows that JB, %!, JB', all lead to the same result, 
viz. A+B=0, which it was our object to prove. 
4 z 
MDCCCLXV. 
