12] 
ON THE THEORY OF DETERMINANTS. 
77 
of determinants, according as we consider the symbol (4) or the more general one (8). 
And this may be done in n or (n — x) different ways respectively. 
It may be remarked, that the symbol (8) is the same in form as if a single 
column only had the mark (*|*) over it; the number n being at the same time 
reduced from w to (w—ar+1): for the marked columns of symbols may be replaced 
by a single marked column of new symbols. Hence, without loss of generality, the 
theorems which follow may be stated with reference to a single marked column only. 
Suppose the letters 
Pu p2,‘“pk', <r u a 2 ,...a k ]&c (14) 
denote certain permutations of 
®1> ®2> • • • > ftl) ($2, • • • ftk 5 &C . (15), 
in such a manner that 
Pi ~ a ffi> P* ~ > • • • Pk — a g k j a 'i = fih 1 ) 1 ••• Gr k = fth k (16). 
Then the two following theorems may be proved : 
t + 
Ap!<r 1... (n)) = ±g ± h ... f Act.fiJ... (n) 
pk&k J 
if n be even : but in the contrary case 
t 
f ••• ( n ) 1 
i pk°k J 
(17), 
h ± g ••• f Aafii •••(n) 
vfik 
(18). 
By means of these, and the equation (11), a fundamental property of the symbol 
(3) may be demonstrated. We have 
t 
which when n is even, reduces itself by (17) to 
Aafi x ... (n)} = 7 ... (ft)'J %(± g ± g . 1) 
afik ) l a kPk J 
+ 
= 1.2 ... k ( Aafii... (n) 
afik 
(20)