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

CHAPTER XXI.

ZERO-AXIAL DETERMINANTS, UP TO 1888.
WHAT was said in the introduction to the preceng htr egding chapter regarding
classification applies also in part here. In addition, it has to be
noted that as in the next twenty-year period there is only one paper
concerned with zero-axial determinants, it is thought best to append
the report of it to the present chapter.
BALTZER, R. (1870).
[Theorie und Anwendung der Determinanten,.... 3te verbesserte Aufi..... viii+242 pp. Leipzig.]
In ~ 4. 2 (p. 29) Baltzer considers the question of the number
of terms in the final development of an n-line determinant having
all the elements of the diagonal equal to 0. Calling the number
i (n), he obtains the correct result
+(X) -- n!1 —1+2T-3~+... +( — l)', ((a)
although making two oversights in the reasoning, and thence
deduces Stockwell's incidental result of 1860 (see p. 3-above),
+(n+1) = (n+l) (n) + ( —l)fl, (3)
the values of /r(1), Vr(2),... being thus 0, 1, 2, 9, 44, 265,....
By using Cayley's development of 1847 he also obtains Stockwell's
identity
n = I/ (n) + n/ (n —1) + I-n(n- 1), (n-2) +... + 1.
It is important to note that the problem here dealt with is identical
with a much older one regarding arrangements, namely, the fidindg'