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

448 HISTORY OF THE THEORY OF DETERMINANTS

The first result follows from diminishing each row in backward order
by the row immediately preceding, and the second from doing this
repeatedly and then treating the columns in like fashion. The former
may also be viewed as a corollary to a theorem of Stern's (1865)
in regard to alternants, and the latter is a simplification of an
instance given (1861) by Hankel in illustration of his theorem
regarding persymmetric determinants, namely that, for example,
when c = 3 and m = 2,
P(10, 15, 21, 28, 36) = P(10, 5, 1, 0, 0).
ZEIPEL, V. v. (1865).
[Om Determinanter, hvars elementer aro Binomialkoefficienter.
Lunds Univ. Arsskrift, ii. pp. 1-68.]
The main subject of this painstaking and methodical investigation
would be more definitely described as being various cases of the
determinant
| (n)m,( + d ()(m+ 2d)p+2e... (i +r ) l.e |,
that is to say, the determinant which not only has for its elements
numbers of the type
r(r-1)... (r-s+l)/s!,
but which has the bases of these numbers identical throughout
one and the same row and regularly increasing throughout each
column, and has the suffixes identical throughout one and the same
column and regularly increasing throughout each row; for example,
(5) (5) (5)2 4 6 4
(6)0 (6)1 (6)2 6 15 20
(7)0 (7)h (7)2, 8 28 56.
A beginning is made with the case
d = 1, p = 0, e= 1,
when it is readily shown by reduction of the columns in order that
the value of the determinant is the same as if m were 0, and therefore is equal to 1. Passing to the case where d 1, e = 1 and p
is not specialized, Zeipel removes from the rows the factors
am l 1, m+1, m+2,...., m+r,