236 
ON THE SEXTACTIC POINTS OF A PLANE CURVE. 
[3 41 
and thence 
(40 x 2 0 3 + 3№) U = 1)3 {(90m - 21) Hd<$> + (- m + 9) <&dH] 
+ (^V)» (_4(3 - v) - ff); 
and adding this to the foregoing expression for (0/ + 100/03 + 60/0 3 + 120J03 2 ) U, 
we have 
(0 1 5 + 100/03 + 100/03 + lo0 1 0 2 2 ) U = 
^2 
^ ni _ ^4 {(27m 2 — 96m + 81)Hd^> + (17m 2 — 56m+51) <£>0/7} 
+ {(- !4m + 22) (0 . V ) H + (- 10m +18) 0 V . 
+ 4 
+ r—-, , 4 dn. 
(m — l) 4 
32. Calculation of 
We have 
0 4 UB 1 H + 203 Ud. 2 H + 2dMd s H. 
0 4 77= 
*y 2 
m— 1 m—1 
021 tf=r^TS H > 
(m—l) 2 
0^ = 0tf, 
d,H=d 2 H, 
d s H = — 1 - (- 3m + 6) 0<ï> - <S>dH + —^- (0. V ) if, 
m — 1 m — 1 v 7 
for which values see Appendix, No. 58. And hence the expression sought for is 
A 2 
(m — l) 3 
m 
1) (§03ff + 0 2 i/) - H^-pVH) dH 
which is 
*y 2 
(m — l) 3 
+ 2 (m — 1) dH d,H 
+ ZH ((- 3m + 6) Hd<& - <$>dH + * (0. V ) H)J , 
f (m-l)0tf 0 2 # 
+ (m — 1 ) dH d n -H 
+ (- 6m + 12) # 2 0<1> - 3H<£dH) 
+ (+ny, ! 2 # (a. v ) h - v if).