224 
ON THE SEXTACTIC POINTS OF A PLANE CURVE. 
[341 
and in addition to the equations, (P = acc + by + cz), 
— (m — 2) 0, 2 U+ P. ^0, 2 P=0, 
— J [(m-l)d 1 3 + 3(m-2)d 1 d 2 ]U+P.%(d 1 3 +3d 1 d 2 )U+d 1 P.±d 1 2 U=0, 
— ^ [(m — 1) (0, 4 + 60, 2 0 2 ) 4- (m — 2) (40,0 3 + 30 2 2 )] U 
+ P. £(0, 4 + 60, 2 0 2 + 40A 4- 30 2 2 ) P + 0,P. £ (0, 3 + 30,0a) P + 10 2 P. ^0, 2 P = 0, 
giving in the first instance 
P — 2 (m — 2), 
0,P = 
<^P 
a 0, 2 P’ 
a D 1 (Si 4 + 60, 2 0o) U d 0, 3 P (0, 3 4- 30,0 2 ) P 
- ~ 2 0, 2 P "0, 2 P 0, 2 P 
and leading ultimately to the before-mentioned value-of II, we have the new equation 
_ a_ [( m - 1) (d* + 1O0, 3 0 2 + 100, 2 0 3 4-150,0 2 2 ) 4- (m - 2) (50,0 4 4- 1O0 2 0 3 )] U 
+ P. T i_(0/ + 1O0, 3 0 2 4- 1O0, 2 0 3 4- 150,0 2 2 4-50A4- 1O0 2 0 3 ) U 
4- 0iP. Yi (Si 4 4- 60, 2 0 2 + 40,0 3 4- 30v) U 
+ |0 2 P . £ (0, 3 + 30,0 2 ) TJ 
4-¿0 3 P • -I b 2 U — 0. 
5. This may be written in the form 
2 [(m - 1) (0,° 4- 1O0, 3 0 2 4- 100, 2 0 3 + 150,0 2 2 ) + (m - 2) (50,0 4 + 1O0 2 0 3 )] U 
+ P{ 
0, 5 4-100, 3 0 2 4- 1O0, 2 0 3 4- 150,0 2 2 
4-50,0 4 4-1O0 2 0 3 ) P 
+ 50,P( 
0, 4 4- 60, 2 0 2 4- 40,0 3 4- 30 2 2 ) P 
4- 1O0 2 P ( 
0, 3 + 30,0 2 ) P 
4- 1O0 3 P ( 
0, 2 P) = 0; 
or putting for P its value, = 2 (m — 2), the equation becomes 
2 (0, 5 + 100, 3 0 2 4- 100, 2 0 3 4- 150,0 2 2 ) U 
4- 50,P (0, 4 -f 60, J 0 2 4- 40,03 4- 30 2 - ) II 
4- 1O0 2 P (0, 3 4- 30,0 2 ) U 
4- 1O0 3 P. 0 2 2 P = 0 ; 
or as this may also be written, 
2 (0, 5 4- 1O0, 3 0 2 4- 1O0, 2 0 3 4- 150,0 2 2 ) P 
4- 50,P. 0 4 P-1- 1O0 2 P. 0 3 P4- 1O0 3 P. 0 2 P = 0.