PROFESSOR CAYLEY ON THE SEXTACTIC POINTS OE A PLANE CURVE. 557 
whence also 
Jac. (U, H, Oh) = 0; 
and the condition for a sextactic point assumes the more simple form, 
Jac. (U, H, ¥) = 0. 
29. Now (former memoir, No. 32) we have 
*=(& 33, C, f, <3, B„H, B~H) 2 
= (l-j-8^ 3 ) 2 (y z z 3 +z z x z +x z y z ) 
+(- 9 n (x*+y>+zj 
5^—20^) (x z +y z +z z )xyz 
+(— 15Z 2 — 78Z 5 +12 l a )xyz\ 
or observing that and xyz , and therefore the last three lines of the expression 
of 'P are functions of U {=x z -\-y z +z z -\-§lxyz) and H(= — l z (x z -\-y ZJ rz z )-\-(l-\-2l z )xyz), 
and consequently give rise to the term=0 in Jac. (U, H, 'P), we may write 
*=(1 + 8 l z ) 2 (y z z z +z z x z +x 3 y 3 ). 
30. We have then, disregarding a constant factor, 
Jac. (U, H, SP)= Jac. (x 3 - i r y z -\-z z , xyz, y z z z -\-z z x i -\-x z y z ) 
= * 2 > y\ * 2 
yz, zx, xy 
^(tf+z 3 ), y%z'+x 3 ), z%^+y 3 ) 
= (y’—z‘)(z’'-x‘)(x’—y‘), 
so that the sextactic points are the intersections of the curve 
TJ=ix z -\-y z +z z +6lxyz=0, 
with the curve 
Article Nos. 31 to 33 . — Proof of identities for the first transformation. 
31. Calculation of (5J+105 3 5 2 +10^B 3 +15B 1 5 2 )U. 
Writing B in place of D, we have (former memoir, No. 20) 
But 
former memoir, 
Nos. 21 & 22 ; 
— B 2 H = ( 3m 6 )( ^_7) ho _ 
(m-l ) 2 
— B 2 H = 
