PROFESSOR CAYLEY ON THE SEXTACTIC POINTS OF A PLANE CURVE. 575 
and thence that 
dH= 
d«, d^, d</ 
+ 
«, K g 
+ 
«, h , g 
K b, / 
Zh, Zb, Zf 
h, i , f 
/ c 
g, /, . * 
Zg , d/, dc 
=(sr, fc, «xa«, ty)+®, as, zb, y)+(®, jr, y d C ) 
=(9, . . .JZa, Zb, Zc, 2Zf, 2 Zg, 2Zh, 
we see that 
Moreover 
and thence 
that is 
Hence the equation 
becomes 
that is, 
□ dU=dH. 
ru= (a, ...JiZ,-i*d., ...) 2 U 
= a(bv 2 -\-cp?—2f(v » ) 
-f J(cX 2 +«i/ 2 —2gvk ) 
+ c(a[jj 2 + hk 2 — 2 likfjb) 
+ 2 /*( —f x 2 -\-gX(jj + AXf — ap ) 
+ /V ~ 9P* + % - ^ ) 
+ 27i( /A -\-gvgj—hv 2 — cX[a ) ; 
rdu= :.,)-du 
= a^dS -J- c? — 2 'fAvZf) 
+&c. 
= JV 2 (6dc+cd5 -2/d/) 
+&c. 
=(*a, dB, bc, zf, d®, d^xx, ^ ,) 2 , 
rdu=dd>. 
<bndU-V 2 dU=HrdU 
<bdH- V 2 dU=Hdd>, 
V 2 dU=d>dH-Hd$. 
69. Proof of equation d 1 d 2 U=-7^ : Yy 2 (OdH— IidO). 
We have 
^2 = (m- i]2^ 2 (^d, r +yd y + zZ z y 
— (to- i)2 ^(^df+ydj+sdJV 
_i_— — V 2 - 
T (m-l ) 2 ’ 
