t 
78 
ON THE THEORY OF DETERMINANTS. 
[12 
But when n is odd, from the equation (18), 
(Aai, &...(»)') = fia 1 jS 1 ...(»))S(±jl) = 0. 
a k> fik 
since the number of negative and positive values of ± g are equal. 
■(21), 
From the equation (20), it follows that when n is even, the value of a symbol 
of the form 
t 
(Aa^i, (w)l 
a k(3k 
(22) 
is the same, over whichever of the columns a, /3... the mark (f) is placed, 
denote this indifference, the preceding quantity is better represented by 
( Aa lt /3, ... (n) 1 
i [ (23), 
( &kfik J 
To 
this last form being never employed when n is odd, in which case the same property 
does not hold. Hence also an ordinary determinant is represented by 
t 
t 
(HI 1 
Vkfik. 
kk J 
(24), 
the latter form being obviously equally general with the former one. 
It is obvious from the equations (17), (18), that the expression (22) vanishes, in 
the case of n even whenever any two of the symbols a are equivalent, or any two 
of the symbols (3, &c.; but if n be odd, this property holds for the symbols ¡3, &c., 
but not for the marked ones a. In fact, the interchange of the two equal symbols, 
in each case, changes the sign of the expression (22), but they evidently leave it 
unaltered, i.e. the quantity in question must be zero. 
Consider now the symbol 
( (25), 
( k k j 
which, for shortness, may be denoted by