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

468 HISTORY OF THE THEORY OF DETERMINANTS

arrangements of the counters, 1, 2,..., n, subject to the conditions that
in every case the kth counter shall occupy neither the kth place from
the beginning nor the k"V place from the end, and determinants are
not in any way referred to. His final result is the recurrenceformula
(2 (n — 2) 'v,,_3 for n even,
=2 (n-1) w-_2 for n odd;
so that, since w2 = 0, w3 = 0, w4 = 4, he finds
W5 = 16, w6 = 80, V7 = 672, w8 = 4752,.
HANSTED, B. (1880).
[Trois theoremes relatifs a la theorie des nombres. Journ. de sci.
math. e astron., ii. pp. 154-164.]
The second theorem established (pp. 156-158) is that the number
of terms in an n-line zero-axial determinant is the nearest integer
to n! e-.
SZUTS, N. v. (1888).
[Zur Theorie der Determinanten. Math. Annalen, xxxiii.
pp. 477-492.]
The chief object of the author here is to generalize Weyrauch's
result of 1871, and this with a wealth of formulae he fully effects.
He is unaware, however, of Cunningham's paper of 1874 and Dickson's
of 1878. The way in which he formulates their and his principal
result is: The number of non-zero terms in an n-line determinant
having r zeros in its main diagonal is the (n-r+l)t" member of the
]th row of differences of
1, 1!, 21,,, 4!....