26 
Transactions of the Royal Society of South Africa. 
I {a\ — a^y {ai — a^y 1 
(ao-a^y 1 
0-1 — a.i ai 
1 
otg — di . a^ — a.^ 1 
— ai a.^ — CL-> ' 1 
-1 -l'^ -1 . 
(11) In connection with §4 we had occasion to note that, when the 
permanents to which we were there led were of odd order, they had the 
value 0. On investigation it was found to be generally true that zero-axial 
skew permanents of odd order vanish. Probably the simplest proof is grada- 
tional. Thus, the theorem manifestly holding when the order is the 3rd, let 
us examine the case of the 5th order, say the permanent 
+ + 
a 
h c 
d 
— a 
e f 
9 
-h 
— e 
h 
i 
— c 
-f 
-h 
j 
-d 
-9 
-i -j 
Taking binary products of the elements of the first row and the first column 
we see at once that the cof actors of — a^, — 6'^, — c^, — are all 0, and that 
the cof actor of any other product is the sum of two three-line permanents one 
of which is the product of the other by (— 1)^ and thus cancels that other. 
(12) As for zero-axial skew permanents of even order our only incidental 
result concerns the bordering of 
+ + 
a he 
— a . d e 
-d . f 
— c —e -f . 
whose value is 
or 
a^f^ + I'ie'i + + 2afhe - 2afcd 2becd 
0 c 
d e 
f 
2 4- 4 I a/ cd he 
cd af 
namely, the common element of the bordering lines being 0, 
""^^^ ^(^fl^,eM,dL,cJj,hM,aN^ 
I 
m 
n 
r 
a 
a 
h 
c 
/3 
— a 
d 
e 
y 
-d 
f 
8 
— c 
— e 
-f 
Imnr 
where 
Jj =z — af + he -\- cd 
M = af he cd 
N = af he cd. 
•Cape Town, S.A. ; December 28, 1915. 
