•341] ON THE SEXTACTIC POINTS OF A PLANE CURVE. 253 
•or since 
3. (xd x + yd y + zd z ) = 3, 3A = 0, 
this is 
03 = “ ¿1 80 (xdx + V d » + zdz > ~ + 01 V ‘ 
67. Proof of expression for 3 3 H. 
Operating with d 3 upon II, we have at once 
dji = - —~ 6 #3$ —~ $0// + (3.w)H. 
m —1 m — 1 m — 1 v 7 
The remainder of the present Appendix is preliminary, or relating to the in 
vestigation of the expressions for d$fU and 37/, 77, used ante, No. 31. 
68. Proof of equation V 2 377= $3i/— i/3$. 
We have identically 
(2i, ...$>, g, y) 2 (2l, 3 y , 3z) 2 —[(21, g, v§d x , 3 y , 3 Z )] 2 
= (abc — &c.) (a, . ■■\vd y — /xd z , \d z — vd x , gd x — Xd y ) 2 ; 
$□ - V 2 = HT; 
and then multiplying by 3, and with the result operating on 77, we find 
^□3i7- vw=mw. 
that is 
Now 
and thence 
and observing that 
and thence that 
□ 77= (31, ...3a, d y , d z y 77 
= (3l, ...$a, 3, c, 2/ 2g, 2A); 
□377= (31, ...$3a, 33, 3c, 23/, 23g, 23/i); 
/7 = 
«, 
A, 
g 
/0 
3, 
f 
f, 
G 
«, 
/6 
¿7 
+ 
a, 
3/q 
33, 
3/ 
h, 
<7> 
/ 
c 
3g, 
we see that 
3 H = 3«, 3/i, 3/7 -f 
h, b, f 
g> f c 
= (3l, a @3[3a, 3h, 3g) + (& 33, №3, 33, 3/) + (@, 0, (S^, 3/ do), 
= (3i, ...313«, 33, 3c, 23/ 23/7, 23/q 
□377 = 3#.