341] 
ON THE SEXTACTIC POINTS OF A PLANE CURVE. 
227 
d 4 Ud x H + 203 770 2 77 + 2 0 2 770 3 77 
-- {(- 6m 2 + 18??t - 12) 77 2 0<l> + (- 17m 2 + 60m - 55) 77<3>0<1> 
+ {(2m — 2)H (3.V ) H + (8m-16) dHV H\ 
+ (^'- n3H î- 
dJJdJJ = 
A 4 
77077. 
(m — l) 4 
10. And by means of these the condition becomes 
V77 2 
0 = (m—I)« {(153m 2 — 594m + 549) 7704> + (— 102m 2 + 396m + 366) <f>077} 
+ {(-96m +168) 77 (0. V) 77 + (- 90m + 162)770V774- (120m - 240) 077V77} 
+ {977 2 0X2 - 4577X2077 + 40^077}, 
being, as already remarked, of the degree 5 in the arbitrary coefficients (X, \i, v), and 
of the order 12 m —22 in the coordinates (x, y, z). 
11. But throwing out the factor *b 2 , and observing that in the first line the 
quadric functions of m are each a numerical multiple of 51m 2 — 198m +183, the 
condition becomes 
0 = (51m 2 - 198m + 183) 77 2 (3#0$ - 2$077) 
+ A {(- 96m +168) 77 2 (0.V) 77+ (- 90m +162) 77 2 0 V 77 + (120m - 240) 077VH} 
+ ^ 2 {977 2 0X2 - 4577X2077 + 40^077}. 
Article Nos. 12 and 13.—Second transformation. 
12. We effect this by means of the formula 
(m - 2) (3770$ - 2$077) = - A Jac. (77, $, 77), (J) 0) 
for substituting this value of (3770<3>-2$077) the equation becomes divisible by A; 
and dividing out accordingly, the condition becomes 
_ 51m 2 -198m+ 183 ^ Jac ^ ^ ^ 
m — 2 
+ (- 96m + 168) 77 2 (0.V)77+(— 90m + 162) 77 2 0 V 77 + (120m - 240) 77077V77 
+ * (977 2 0X2 - 4577X2077 + 40^077) = 0. 
(J) here and elsewhere refers to the Jacobian Formula, see post, Article Nos. 34 and 35. 
29—2