*252 
ON THE SEXTACTIC POINTS OF A PLANE CURVE. 
[341 
or since 
this is 
VU = -^—H, У 2 £7=ЯФ, 
m— 1 
л „ tj m U , . , 'à 2 rT . , 
371/ = 7 ^ Ф- + Г г 2 ЯФ. 
(m — l) 2 (m — 1)- 
64. We have 0,0 2 Z7 = 0, and thence 
that is 
(№ + 3,03 + 02) U=0, 
0103 U — — 0 1 2 0 2 U — 0 2 2 U ; 
or substituting the values of didJJ and d.?U, we find the value of d-^.U as given in 
the Table. And then from the equation 
(3/ + 6320., + 43,03 + 302 -t- 3 4 ) U = 0, 
or 
d 4 U=- (32 + 63200 + 40,03 + 30 2 2 ) U, 
we find the value of d 4 U, and the proof of the expressions in the Table is thus 
•completed. 
65. Proof of equation V . 3 = 0. 
We have 
V . 3 = V . ((Bv — С/л) d x + ((7л, — Av) Эу + (Л/л — BX) d z ) 
= V . (A (jxd z — vd y ) + В (vd x — Л2 г ) + С (Хд у — gd x )) 
= V A (ftd z — vdy) + V В (vd x — Xd z ) + V С (X8 y — цд х ); 
nnd then 
VA=(2l, ...$\ /л, v][a, h, g) = HX, 
vs= (21, /a, v$h, h, f) = Hfi, 
VG = (21, g, v\g, /, c) = Hv; 
or substituting these values, we have the equation in question. 
66. Proof of the expression for 3 3 . 
We have 
d* = - w _ ! Ф + Уду + *9*) + j V; 
■and thence operating on the two sides respectively with 3,, =3, we have 
0 3 = - ?j 22Ti + y^y + zd z ) + ФЗ . (жд х + уду + zd z )} 
+ —Ц- ШУ + ^3. V}; 
in — 1