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

ALTERNANTS (BETTI, 1857)

183

observation of his own, namely, that the said symmetric function
is likewise the coefficient of t-(al+l)t-f(a2+l). t-(a+l) in the
similar development of.fW - (...2 * (al 2) ]. 2 (t ), ).
f'(tl).f (t2)... f(tn) * 2(t1, t2..., tn)
and by comparison of the two results draws the conclusion that
if Borchardt's generating function be denoted by
0(tl, t2.., tn),
and his own after removal of II2(x1, 2,..., x,) from the
denominator be denoted by
O(tj, t..., tn)
the squared difference-product of the x's is equal to
"95(tll t~ t-n ( l+ l ) -(a2+l)
1 (2 * '' n
where the notation used is sufficiently explained by saying that
in accordance with it the coefficient of x' in the expansion of
F (x) is denoted by
{F(X)},.
BALTZER, R. (1857).
[THEORIE UND ANWENDUNG DER DETERMINANTEN,........
vi+ 129 pp., Leipzig.]
The section ( 12) dealing with the "Product aller Differenzen
von gegebenen Grossen" belongs to the second part of Baltzer's
text-book, that is to say, the part concerning "applications." It
occupies eleven pages, those devoted strictly to alternants being
the first three (pp. 50-53).
At the outset he establishes the determinant form for
the difference-product P(al, a2,..., a): then he gives two