686 Proceedings of the Royal Society of Edinburgh. [Sess. 
where rst ... is any set of n — 2 integers taken from 2, 3, ... , n, and a is 
the one remaining. The term therefore is 
-L | [12]' 2 [23 . . . n] 2 + [13j-’[324 . . . nf + 
t 
i.e. 
i [23 . . . ?if . | a l2 ' 2 + « 13 2 + . . . + a ln 2 | . 
Thus, taking n = 7 we have 
- A 7 = (¥!> 2 + ••• + } 
+ aj| [[12][2567] + [13][3567] + [14][4567]y J . . . 1 
+ —[23 . . . 7 + a 13 + ... + a,--) . 
X 
From (xxxviii.) it follows directly that If | a 11 a 22 . . . a nn |, or A say, be 
a skew determinant with nnivarial diagonal, then when n is odd A n . r x r x 
and aA are both positive, and when n is even — A 11 .r 1 r 1 and A are both 
'positive, the former in each case being the greater . (xxxix.) 
( Issued separately October 15 , 1909 .) 
