48 NOTE ON THE THEORY OF DETERMINANTS. [309 
and prefixing the sign (+ or —) of the arrangement; and the resulting arrangements, 
for instance 
+1 
1, 
-12,- 
-1 2, 
2 
2 
2 1 
2 3 
3 
3 
.3 3 
3 4 
4 
4 
4 4 
4 1 
are interpreted either into +11 . 
22. 
33 
.44, -12 
.21.33.44, -12.23.34.41, or in the 
notation of the formula, into 
+ 1 1 1 2 | 3 | 
H 
i 
■ 1 12 | 3 | 
1 4 |, - | 1 2 3 4 |; 
and so in general. 
Suppose that any partition 
of 
n 
contains 
a compartments each of a symbols, 
fi compartments each of b symbols ...(a, b,... being all of them different and greater 
than unity), and p compartments each of a single symbol, we have 
n = aa + fib + ... + p ; 
and writing, as usual, Ila = 1.2.3 ... a, &c., the number of ways in which the symbols 
1, 2,. 3, ...7i, can be so arranged in compartments is 
Tin 
(n ay {my ...n«n/S...n P ; 
but each such arrangement gives (ll (a — l)) a . (U (b — 1)Y terms of the determinant, 
and the corresponding number of terms therefore is 
Un 
a a W ... 11a 11/3 ... Tip' 
The whole number of terms of the determinant is Tin, and we have thus the theorem 
l-S 1 
... Ila II/S ... lip ’ 
in which the summation corresponds to all the different partitions n — m + fib,...+ p, 
where a, b, ... are all of them different and greater than unity; a theorem given in 
Cauchy’s M¿moire sur les Arrangements &c., 1844. But it is to be noticed also that, 
the number of the positive and negative terms being equal, we have besides 
0 = 2 
\<x [Cl—1) -f ß (b—1) + • 
(-> 
or, what is the same thing, 
0 = 2 
a a b ß ... na 11/3 ... Tip’ 
(_yi-a—ß p 
a a b ß ... Ila n/3 . . np ’ 
* = S 
1 
a a b ß ... na Ufi ... np ’ 
and thence also