76 
ON THE THEORY OF DETERMINANTS. 
[12 
This being premised, consider the symbol 
Ap 1 a 1 ...(n) 
(4), 
Pk a k 
denoting the sum of all the different terms of the form 
ir is**- Ap n a Si A.p rk cTg k (5), 
the letters r 1} r 2 ...r*; s lf s 2 ...s k ; &c (6), 
denoting any permutations whatever, the same or different, of the series of numbers 
(2) [and the several combinations of pa... being understood as denoting suffixes of 
the A’s]. The number of terms represented by the symbol (5) is evidently 
(1.2 ...k) n (7). 
In some cases it will be necessary to leave a certain number of the vertical 
rows p, a ... unpermuted. This will be represented by writing the mark (*f*) immediately 
above the rows in question. So that for instance 
t t 
f A Pl a x ... ^ ... (w)^ 
' : • (8). 
[ Pk<*k • • • @k4>k J 
the number of rows with the (*f*) being x, denotes the sum of the 
(1.2 ... k) n ~ x (9) 
terms, of the form 
±r±s ••• Ap ri a Sl ... 0 1 (f> 1 ... Ap rk a Sk ... 0 k <f) k (10). 
Then it is obvious, that if all the rows have the mark (-f-) the notation (8) denotes 
a single product only, and if the mark (-}*) be placed over all but one of the rows 
the notation (8) belongs to a determinant. It is obvious also that we may write 
t t 
Api&x • ■ ■ 0i(pi • • • (w) | = — i u i v • • • [ Ap^i... 0 Ui (f> Vi ... (w) | 
: ; (11), 
Pk<Tk • • ■ @k<f>k J l P k<Tk • • • J 
where % refers to the different permutations, 
Wx, n 2 ,...u k \ v u v 2 ,...v k ) &c (12), 
which can be formed out of the numbers (2). The equation (11) would still be true, 
if the mark (-f*) were placed over any number of the columns p, a ... 
Suppose in this equation a single column only is left without the mark (-f*) on 
the second side of the equation; the first side is then expressed as the sum of a number 
(1.2... &) n-1 , or generally (1.2... k) n ~ x ~ l (13),