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

COMBINATORY NUMBERS (ZEIPEL, 1865)

4?51.

and
(5)2 (53 (5)4 (5)2 (6)3 (7)4,
(5), (5)2 (53 (6)2 (7)3 (8)4
(5)0 (5), (5)2 (7)2 (8)3 (9)4
are additional equivalent forms.
The next Section (pp. 13-41) of the memoir concerns the cases
where the bases or the suffixes do not proceed by the common
increment 1. The simplest case is where e = 1 and d is not
specialized, the result being
(m)-v (M +d),,, +. (m +rd)p+,r
(~in~rd)p... (m+d)p(m)p dB?."+1
(p+r)p. (P+l)p(P)p
This is obtained (pp. 30-35) by removing factors as in the second
procedure above indicated, and then showing that the determinant
so reduced,
(m-p)0(m-p+d)1... (m-p+rd),.l dt'r+1.
The corresponding case where d = 1 and e is not specialized is
treated in like fashion (pp. 35-41), the product of the removed
factors being
(m)p (m +,..p (m + r)p
(p)p(p+e)p... (p+re),'
and the reduced determinant
(rn-p)0(rn+-p+1)....(a r)7
For the latter there is obtained the lengthy expression
P(mn-p, r) ~ P(rm-p- e+ 1, r —1)... P(mn-p-'r —1. e- 1, 1)
where P(rn, r) (rnerm1e1.
On changing the left-hand member so as to have the base of the
elements constant, and taking p = 1, e - 2, r = m-i, the outcome is
(n) 0 (rn)2 (r) 4.... 2P
(rn1 (M) 3....
rn-i