PEOEESSOB CAYLEY ON CUBIC SUEEACES. 
267: 
67. This is 
(kwf. 6 
+(7^) 5 . -^ 2 - 6^+w 2 
+ (kwy. <7 2 .30 + (7+2y++. 67 + 127 2 0-47y 2 -1440y 
_|_ (jc W ) 3 . _ 2 ff ^ + ff 2 (2y 2 -1074)+ o+( - 8tv - 1 444) + +( - 1 27 2 + 36y) 
- 8f0 + 47V + 2887y0- 32y 3 — 17284 
+•( lew ) 2 . ff 4 30+<7 3 +4y+(7 2 + ! .127+cr+ J . 37 
+<r 2 (127 2 S-47y 2 -1444;)+<r^( + 87 2 y + 28870-96y 2 )+4/ 2 (87 3 -727y+8644) 
-f -(Jew ) . — ffS7 2 +<7 4 ( — 67$+y 2 )+<7 3 \f/( — 87y — 1440) 
+ff ^(_87_90y)+<r^ 3 .~72+4/ 4 . -108 
+W^. 2y > 2#» ^) 3 =0, 
which, reducing the last term, is 
{Jcwflmnxyz 
— 4(7 3 kpv(y — z)(z — x)(x—y)(ny — ?na)( Iz — nx)(mx —ly)— 0. 
68. I verify the last term in the particular case 2=0 as follows : the coefficient of <r 3 is 
( 0 , 2n(l+7ii)xy, 2m-\-nx-\-2n-\-ly, 4^J,x-\-yy, nvxy ) 3 , 
which is 
= i ln 2 vx 2 y 1 { (7 + m)(A# + 
+ (m + w# + n + ly)(v'KX + vyy) 
+2 mfxy] 
= 2ifvx‘Y { 7 + m/. + m + m)\x* 
+ [ 27 + mXyj + v(m + ny, + n + lx + 2 nv)~\xy 
+(7+m/a+w+7j') i a# 2 }, 
which, substituting for X, yu, v their values m—n, n—l, l—m , is 
= 2 w 2 j >x 2 y 2 . — Zkyj(x—y) (mx — ly) ; 
or for 2=0 the coefficient of <r 3 is 
= —4 xyjm 2 x 2 y 2 (x—y)(mx—ly), 
agreeing with the general value 
— 4 xyjv{y — z) {z — a?)(# —y)(ny — mz)(lz—nx)(lx — my). 
2 o 2 
