309]
NOTE ON THE THEORY OF DETERMINANTS.
49
where, as before, n = act + /36 ... + p (a, b,... being all different and greater than unity);
but the summation is restricted either to the partitions for which n — cl — (3... — p is
even, or else to those for which n — cl — /3 ... — p is odd.
The formula affords a proof of the fundamental property of skew symmetrical
determinants. In such a determinant 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... n) = | 1 | 2 | ... | n |
+ &c.
of the determinant, there is at least one compartment | 1 | or | 123 j &c. containing
an odd number of symbols : let | 123 | be such a compartment, then the determinant
contains the terms | 123 | P and | 132 [ P (where P represents the remaining com
partments), that is, 12.23.31. P and 13.32.21. P. But in virtue of the relations
12 = — 21, &c., we have
12.23.31 =-13.32.21;
and so in all similar cases; that is, the terms destroy each other, or the skew
symmetrical determinant of an odd order is equal to zero.
The like considerations show that a skew symmetrical determinant of an even
order is a perfect square. In fact, considering for greater simplicity the case n = 4, any
line in the foregoing expression of (1234} 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} = + | 12 | 34 | 3 terms
— | 12 34 | 6 terms,
which is in fact the square of
12.34 + 13.42 + 14.23;
for the square of a term, say 12.34, is 12 2 .34 2 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 2.12.34.13.42,
or -12.24.43.31 -13.34.42.21, that is - | 1243 | - | 1342 |, and so for the other
similar terms, and we have
{1234} = (12.34 + 13.42 + 14.23) 2 :
and so in general, n being any even number, the skew symmetrical determinant
{123...w} is equal to the square of the Pfaffian 12 ...n, where the law of these Pfaffian
functions is
1234 =12.34 +13.42 +14.23
123456 = 12.3456 + 13.4562 + 14.5623 + 15.6234 + 16.2345,
where, in the second equation, 3456, &c. are Pfaffians, viz. 3456 = 34.56 + 35.64 + 36.45 ;
and so on.
2, Stone Buildings, W.G., December 28, 1860.
C. Y.
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