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

52 HISTORY OF THE THEORY OF DETERMINANTS

SAGAJLO, A. (1874).
[WYKLAD ZUPEINY ALGEBRY.... Tom ii. Teorya Wyznacznikow
i ich przedniejsze zastowania.... xii+400 pp. Paryz.]
The volume, although like others entitled 'The Theory of Determinants and its principal applications,' is practically a Polish
edition of Salmon's Modern Higher Algebra, with a few additions
and amendments.
ALBEGGIANI, M. (1874).
[Sviluppo di un determinante ad elementi polinomi. Giornale di
Mat., xiii. pp. 1-32.]
Albeggiani begins by restating his development of 1872 for the
case of binomial elements, and then he proceeds to establish it for
the case of trinomial elements which he states in the form.(n)~, (0n) (,] (n1)
(3) = 1 + +A3
+ -A1AA1 A 1 3 +....
+ A (it-^ h)A(1 ) + - A((, - k)A ()
+ E 2A (1- ) A (1) A()
+4- EA1 2,
2+ 2A Ag g3 +.
where the minors following any 2 are quasi-complementary,-that
is to say, no two of them have a row-number or a column-number
in common, and yet all the row-numbers are used. This naturally
suggests the statement which is likely to hold for all cases; and
on the hypothesis that it is true for the case of p-termed elements
he sets about trying to prove it true for the next higher case. With
exemplary patience he gives seven pages to the attempt and is
successful, the final result thus being in double-I notation
(i) ( / \ (a) A() (7r)
w ~he~re= a +- +... + r ),=
where a+f3+... +7r.