642 
MR. A. CAYLEY’S THIRD MEMOIR UPON QUANTICS. 
No. 64. 
There is an octicovariant for which we may take 
©u= 
S.HU, 
S,HU, 
B.HU, 
u. 
U, 
S HU, 
m u 
or else 
©,u= 
iB. u, 
R U, 
A u. 
HU, 
BRHU, 
HU, 
b.b^hu. 
B.B.HU, 
B^HU, 
or else, what I believe is more simple, a function 0;;U, which is a linear function of 
the last-mentioned two functions. 
The relations between ©U, ©,U, ©^^U are — 
-©,U-|-4©U=T.U^-24S.U.HU 
©,,U-1-2©U=T.U^-10S.U.HU. 
I have not worked out the developed expressions. 
No. 65. 
The cubicontra variant is PU= 
r 
-1 bcl 
— 1 acl 
— 1 abl 
+ 1 ack 
+ 1 abi 
+ 1 bcj 
+ 1 abg 
+ 1 bch 
+ 1 acf 
— 1 abe 
r\ 
+ 1 bffi 
+ 1 agi 
+ 1 afk 
-2 afg 
-1 af 
-1 6/ 
+ 1 afl 
+ 1 bgl 
— 1 aP 
+ 1 afi 
+ 1 cf/e 
+ 1 chj 
+ 1 bhj 
+ 1 ail 
— 2 bgh 
-2 cfh 
— 2 aik 
— 2 bij 
"h 1 chi 
-t-1 % 
—1 fy 
— 1 g-h 
— 1 fK- 
-1 ch^ 
+ 1 bjl 
+ 1 ckl 
-1 6/ 
-1 c/r 
— 2 cjk 
+ 1 chk 
+ I fii 
+ 1 gji 
1 ckL 
+ 2^-^ 
+ 3 fhl 
+ 3 fgl 
-1 
+2/!/- 
-1 M 
+ 3 fgk 
*N 
— 1 ?:Vi: 
— 1 if 
-1 jk~ 
+ 3 fhl 
-1 fv 
— 1 ghk 
-1 
- 1-2 g^^k 
i 
-1 gjk 
+ 2 gk- 
— 1 gki 
+ 2 kH 
— 1 //^^• 
— 1 ghi 
— 4 gkl 
— 1 hij 
— 1 hki 
+ 2 hi^ 
-2 hP 
-2fP 
-2 gP 
— 4 Ml 
-2 jP 
-2 kl^ 
-2 il“ 
+ 3 
-|- 3 ikl 
3 ijl 
+ 3 ijk 
+ 4 P 
J 
