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

ZERO-AXIAL DETERMINANTS (MUNRO, 1872)

465

MONRO, C. J. (1872).
[Baltzer on the number of terms in a determinant with a vanishing
diagonal. Messenger of Math., ii. pp. 38-39.]
Besides pointing out Baltzer's faulty reasoning, Monro provides
a substitute. Expressing a zero-axial determinant of the (n+1 )th
order in terms of the elements of the first row and their cofactors,
he obtains at once
(n + 1) = n{N) A-+fr(n - 1)}.
He then writes this in the form
-r(n + l) - (n+) = -VBk(Xn)(-(n)- (n 1)
which leads finally to
V1' (n+ 1) - (n+ 1) ~frlu) = (- 1)~1L-'fr(2)- 24,(l)]
)- n-I [I 0
WEIHRAUCH, K. (1874).
[Zur Determinantenlehre. Zeitschrift f. Math. u. Phys., xix.
pp. 354-360.]
After an introduction and a correction of Baltzer,-who had
corrected himself the year before,*-Weihrauch gives two proofs
of the result
neither of them being short or attractive. Using the longer recurrence-formula, he obtains of course
f(n) = (Jl)n (n) 1.
(n)2z (n-i) 1>
(2),2-1 ('a-l).2 (n - 2)-:3....
(n2)n (n - 1), (qt- )( (n - 1) 1 0,z - 22) 0..!0
and this is his starting-point in both cases.
B Jerichte... Ces. d. Wiss. (Leipzig), xxv. pp. 523-537.

M.D. III.