34 
Transactions of the Royal Society of South Africa. 
n \ {aia.^...an + h^\...hyi} 
+ 1) ! 1! {^a,a^...a,,^i^h^ + SaiSM2---^M- 
H- i^n - 2) ! 2 ! {S%«'2---«»-2S&A -r Sotia^^^A---^^-!} 
H- 
the number of such double terms being \n when n is even and \{n + 1) 
when n is odd, but there being also in the former case a unique single 
term. 
(8) If now in this we put — — ar and take n — 2m, the left-hand 
member becomes a zero-axial skew permanent, and the right-hand member 
becomes 
2- (2m) ! aia2...a2m 
— 2- (2m — 1) ! '2.aia.,...a2m-i ■ Sai 
-f 2-(2m — 2) ! 2 ! Sai«2...a2,„_2 • 
+ ( — 1 ) m ! m ! ( ^a^a.^ . . . ) 
. 2-(2m) ! C2rn 
— 2-(2wi — 1) ! C2,„.-iCi 
or + 2-(2m — 2)! 2! C2,„_2C2 
+ (- irimlfcl. 
(9) When is 2 this expansion is 
2-4 2-3!c3Ci + (S!)^/^^ 
^. e. 4(c2 - 3ciC3 -f 120^) ; 
when m is 3 it is 
2-6 ! Cq - 2-5 ! cjc'i -r 2-4 ! 2 ! c^Co - (3 ! 
^. e. - 12(3c3 - 8C0C4 + 2OC1C3 - 120c6) ; 
and when m is 4 it is 
288 (2c^ - 5C3C- -f lOcoCg - 35C1C7 2800^); 
where the bracketed expressions are exactly those at the end of §6. We are- 
thus face to face with the curious identities 
10 
35 
hci Co 
5co Co 
6c^-CiCs 
- ^7c, 
-7cci C3 
8C6-C1C5 
a^ — ao — ^3 a^ — a. 
ary — a 
a.T — ao ac — a* 
^3 — a^ a3 — tt o . 0,3 — a J 
+ 
ai — ^2 — . . . — O-g 
a.T — cti 
ao — tt o ... etc, — 
a^j — ai a ' — tto 
3~% 
■^3 — «6 
tfft — 0-1 — ao a^ — ao... 
