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

480 HISTORY OF THE THEORY OF DETERMINANTS

Similarly, we are told, in the case/ of the fifth order there is
obtained

1 I I 1 1.
1 e e2 e3 4
1 62 64 e6 6 8
1 C3 C6 C9 e12
1 e4 68 612 16

1
1
or I
1
1

1 1 1 1
e e2 e3 64
62 64 3 63
63 e 4 2
64 63 62 e

where e is a primitive fifth root of 1, and the value of the determinant is (55)1. It is added that this holds generally when the
order-number & is a prime, the value being then
( _ L i)2'(2n-i ))(n-) 1,2.
When the order is composite, the results are not so simple, the
variety introduced being due to the different modes of resolving n
into factors. The case where n is 4 is dealt with in detail.
VELTMANN, W. (1871).
[Beitrage zur Theorie der Determinanten. Zeitschrift f. Math. u.
Phys., xvi. pp. 516-525.]
The third of Veltmann's contributions is in effect the identity

X (t1
x a1
t1 x2. a2
aC1
a1 t2
a. a2^

a,,..... n-1 1
x-),.... at 1 1
X.... anx 1
a3..... x I
3 ' ' ' ' n 1

= (x- t) (x - a2)... (x-a.).

He does not note that the identity obtained by deleting the last
row and column in the left-hand member and substituting
x+al+a2+.. +an-.
for x-a, on the right is much more interesting, being naturally
first in order of thought, and the parent of the other by
'bordering.' The case where the bordering lines cross in a
zero is also worth noting.