138 
Proceedings of Royal Society of Edinburgh. [sess. 
+ 
1 1 a\a\ I | cl\cl\ | | a\a\ I . . . | 
+ + + 
1 1 - 1 1 a\a\ I I a\a\ i | a\a\ I ... I a\lZ\a%T} j 
+ 
or 
+ + 
= XI I a X I I “X I I “X I ... I <‘- s ar~‘ 1 1 , 
where Ilk , pq , rs , . . . , yz is one of the ways of separating the 2 n 
suffixes 1, 2, 3, , 2 n into n sets of 2 each, and where the sign 
of summation implies that all other such separations are to he 
taken, it being understood that the sign preceding any permanent 
is to be made the same as the sign of that particular term of the 
alternant which is brought into prominence by the notation em- 
ployed in specifying the permanent. Since, however, in specifying 
the typical permanent the particular term of the alternant which 
makes its appearance is 
• • • • 
the sign of which is ( - l)g where g is the number of inverted 
pairs in h hp q . . .y z , a better way of writing this deduction from 
the general theorem is 
+ + 
j a*a\ . . . all- 1 \ = 2, ^ 1 1 a i a * 1 1 a l a l ' 1 1 * ■ * 1 1 1 1 , 
or, more at length, 
I a^al . . 
all 1 
N 
II S 
1 a h a k 1 
\«\ 
| a\a k I . . 
• • 1 # 
- 2 c$M\ 
1 1 
1 afh\ I 
1 1 • • 
. . 1 <- 
1 «x 1 
1 afl | 
1 <« 5 S 1 • . 
• • I afi- 
1 <J< 1 
1 a l< l z l 
I«xi- • 
••i a p- 
It is then apparent that on the right the differences a k - a h , a 0 - a 
pi 
a s — a r , 
a y are factors of the 1st, 2nd, 3rd, 
7i th rows 
respectively, and that if we remove them we shall have 
| a°a 2 . . . a% n | = (a q — a p ) ( a s — a r ) . . . ( a z — a y ) 
X 
(a h a k )° (a h a k ) 2 (a h a k f . . 
,.(a h a k y"--' 
(a p a q )° (a p a q ) 2 (a p a q y . . 
(a r a s )° m r a s ) 2 (a r a s f . . 
• • (®a) 2 “' s 
(«A)° (vr (a*®.) 4 • • 
v ( a y a z) 2 m • 
