The theory of determinants in the historical order of development, by Sir Thomas Muir.

44 HISTORY OF THE THEORY OF DETERMINANTS

in the result of his effort. It is strange, for example, to find in the
short opening chapter valuable space given to two unnecessary
though interesting theorems regarding derangements while skew
determinants and other important special forms are nowhere
mentioned. The theorems in question are
1. The total number of derangements in the n! permutations of the
first n integers is (n —l)n n!
2. The total number of derangements in the n permutations got by
permuting the first n integers cyclically is (n- 1) n (n-+ 1).
Save for a very special persymmetric determinant there is
nothing else worth noting.
GUNDELFINGER, S. (1873).
[Ueber einen Satz aus der Determinantentheorie. Zeitschrift f.
Math. u. Phys., xviii. pp. 312-315.]
The theorem in question may be said to concern the ' increment'
produced on the product of two oblong arrays by prefixing to each
a column of units. It being understood that the given array
consists of in rows and n columns, the said increment
| 1 (6 ' (.1 ( an a, 1 a(2... (n a * a a
|1 b, 62,*..bn. 1 31 2.. ( i 3n I - /, b, 6... b, 1 /2 ~... n t
Ilb b.... 1bn 1b b
1 2 c... Cn 1 n1 Y2 * * n 2 *. * n l 1 2 On
I 2. n
l + Eaa l + La/3 l + sC.. Eaa a/3 cay...
l + lba 1 + b/3 1 + by... _ Lba b/ Eb7...
+Eca l+ Ec/3 1 + ecy.. Eca Ec3 E cy
r ~~I Eau lao lay1~
=_. 1 1 1
1 aCa aC/3 C cy...
1 ba b/3 2by..
1 Ica Eci3 Icy..n+l
where Eaa stands for aqi + a,22 +...+ — ann. This agrees with
the result obtained in a different way by Sylvester in 1852 where
n - m-1, and where, therefore, the product of the given arrays