184 Mr, A. Cayley on the Theory of Determinants. 
or, what is the same thing, 
eee amie 
a*b? .. ged G: . Tp 
eat ale 
and thence also ~ 
j 
t= 
a* bP... ilall@... 1p’ 
where, as before, n=aa+ Pb...+p (a, b,... bemg all different 
and greater than unity); but the summation is restricted either 
to the partitions for which n—a—6...—p is even, or else to 
those for which n—a—f6...—p is odd. 
The formula affords a proof of the fundamental property of 
skew symmetrical determmants. In such a determimant we 
have not only 12=—21, &c., but also 11=0, &c. Suppose 
that n, the order of the determinant, is odd; then in each line of 
the expression 
{123...nk=|1]2]...12] 
+ &e, 
of the determinant, there is at least one compartment | 1 ] or 
| 123] &c. containing an odd number of symbols: let | 128 | 
be such a compartment, then the determinant contains the terms 
| 123] P and {182|P (where P represents the remaining 
compartments), that is, 12.238.31.Pand18.82.21.P. But 
in virtue of the relations 12=—21, &c., we have 
19 O83 a Wao ae as 
and so in all similar cases, that is, the terms destroy each other, 
or the skew symmetrical determimant of an odd order is equal 
to zero. 
The like considerations show that a skew symmetrical deter- 
minant of an even order is a perfect square. In fact, consider- 
ing for greater simplicity the case n=4, any line in the foregoing 
expression of 11234! for which a compartment contains an 
odd number of symbols, gives rise to terms which destroy each 
other, and may be omitted. The expression thus reduces itself to 
{1234} = +112] 384] 3 terms 
—| 12 34] 6 terms, 
which is in fact the square of 
12.84413.424 14.28. 
For the square of aterm, say 12.54, is 12”. 342 or 12.21 .34.43, 
that is, ] 12 ]34], and the double of the product of two terms, 
say 12.34 and 13, 42, is 2.12.34.18.42, or —12.24.43.31 
