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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