Mr, A. Cayley on the Theory of Determinants. 183 
and prefixing the sign (+ or —) of the arrangement; and the 
resulting arrangements, for instance 
ol hy seks, .. eh s, 
2 2 al 23 
33 3.3 3 4 
4. 4, 44 4] 
are interpreted either into +11.22.83.44, —12.21.33.44, 
—12.23.34.41, or in the notation of the formula, into 
Pii2isi4i, —|12(814b -— t12341. 
And so in general. 
Suppose that any partition of n contains « compartments each 
of a symbols, 6 compartments each of 6 symbols... (a, b,... 
being all of them different and greater than unity), and p com- 
partments each of a single symbol, we have 
n=aa+PBb+ ...+ . 
And writing, as usual, [Ia=1.2.3...a, &c., the number of 
ways in which the symbols 1, 2,.8,...n, can be so arranged in 
compartments is 
IIn 
(IIa)*(I1d)*...TeII8...1p’ 
but each such arrangeraent gives (II (a—1) )* ; (1I(6—1) )é 
terms of the determinant, and the corresponding number of terms 
therefore is 
IIn 
ob, Alas ..idip: 
The whole number of terms of the determinant is IIn, and we 
have thus the theorem 
1 
Wreriesitis clin Weesekl a 
in which the summation corresponds to all the different partitions 
n=aa+8b...+p, where a, b,... are all of them different and 
ereater than unity ; a theorem given in Cauchy’s Mémoire 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 
ye soe hic 
Wea | aipales SO EOL EE Berea 5 Oe, Sree 
Mem ta lde te 
