341] ON THE SEXTACTIC POINTS OF A PLANE CURVE. 
70. Proof of equation 
dl%U = ((3m ~ 6) НдФ + <- m + 3 > ФЭЯ > + (~Ту t-(S • V) H). 
We have . V ’ 
9 3 = l ~r дф (хд х + уду + zd z ) - —i— ФЭ + -5— a. v, 
m — i m — 1 m — 1 
and thence multiplying by Э д 2 = Э 2 , and operating on 77, 
9 1 2 a 3 t7 = -^f 0Ф0 2 77 i- Ф0 3 77 + — (Э. V) 0 2 77. 
m — 1 m — 1 m — 1 
To reduce (0. V)0 2 77, we have 
0(V0 2 77) = V0 3 77 + (0. V0 2 ) 77, 
= V0 3 77+ [(0. V)0 2 + V (0.0 2 )] 77, 
= V0 3 77 + (0. V)0 2 77 + 2 V00 2 77, 
and since 
02 = ~ m _ i Ф (^0* + 2/0?/ + ^0z) + -"-j v ; 
multiplying by V3, and with the result operating on 77, we obtain 
\/дд,и=-^—^ л Ф\/ди+^~- V 2 077: 
m— 1 m—1 
or since V 0 77 = 0, this is 
V 00., 77 = —V 2 Э 77. 
??г — 1 
Hence 
0(V0 2 77)= V0 3 77+(0. V)0 2 T7+ w -^ V 2 077, 
that is 
(0. V)0 2 77 = 0(V0 2 77)- V0 8 77— V 2 077. 
Substituting this value of (0. V)0 2 77, we find 
0 2 0 77 = - —? 0Ф0 2 Т7 — -L- Ф0377 
n? — 1 m — 1 
+ -^- r (0(V0 2 77)-V0 3 77) 
w — 1 
+ (S?rp- 2VW )- 
the three lines whereof are to be separately further reduced.