Mr. A. Cayley on the Theory of Determinanis, 181 
No. of terms. 
spe 
eps ya eli. .e kD 
=e ge! 2 | | E | 
. 1+1=2 
{123}=+4+ ]1[2]3] . at 1 
—|l ae ee 3 
tt & Bh so sin 2 
o+3=6 
{1234,=4+ ]1)2]3}41.- | 1 
—J1l 2|3]4|. . 6 
eb ss, ot4 | a 
+[1L 2]3.4]}.. . 
—|i 2 3-4}. 6 
124+12= 24 
Oo 
(12345}=+ ]112]314]5]. | 1 
cg SASL A | ot 10 
Sly 2.31 4 [oP sty 20 
eb Sd Pere ee 
Bb l 2 3 4 Salis 30 
aid 2 814-5) % 20 
el 23.4. 5. | 04 
-60+60= 120 
where, as regards the signs, it is to be observed that there is a 
sign — for each compartment] | contaming an even number 
of symbols; thus in the expression for {1234}, the terms 
J 12] 34] have thesign — — = +, and the terms| 123 4} 
the sign —. Or, what comes to the same thing; when 7 is even, 
the sign is + or — according as the number of compartments 
is even or odd; and contrariwise when 7 is odd. As regards 
the remaining part of the expression, this merely exhibits the 
partitions of a set of n things; and the formule for the several 
determinants up to the determinant of a given order are all of 
them obtained by means of the form 
