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

DETERMINANTS IN GENERAL (ALBEGGIANI, 1874) 53
To make the form of development more familiar to the reader,
Albeggiani points out its partial resemblance to that of
((a + 2 + ~ ~ + Cp)n;
and on account of the latter development being better known he
uses it as a mnemonic to recall the former. He thus writes his
own theorem in the ultra-symbolic form
(A) = (A +,+2+ A... +A,)
the implied instruction being that the ntt' power of
Al+A +.. +A\
is to be found in the usual way, and that then every term of it is
to be suitably changed; for example,
_____________! a 7 () () (T)
a!/!.. ~r! AA2... into CnA(2) *;A
aiAIA 2^ A" in* to ~A A^ ~ 2... Ap~
where it has to be noted as another point of analogy that the
number of terms under the E is represented by the coefficient!/a!S3!... vr! which the Z displaces.
Consideration is next given to another ultra-symbolic form and
to certain rather fanciful applications (pp. 13-24); and finally
a re-examination is made (pp. 24-32) of Siacci's theorems and
deductions, some of which had been the original cause of Albeggiani's investigations.
GARBIERI, G. (1874).
[I DETERMINANTI, con numerosi applicazioni. Parte Prima. Utile
agli studiosi di matematica nei primi corsi universitari. xiv+
267 pp. Bologna.]
In bulk this exceeds all previous Italian text-books except
Trudi's (1862); and, as appears from the title, it was intended to
be much larger; the Second Part, however, has not been published.
Notwithstanding its greater fullness and general soundness, it
cannot be said to show a marked advance in any particular. Of its
twenty-two chapters the number strictly devoted to the theory is
less than half, much space being given to geometrical * and alge* Those interested in such applications will find a valuable paper of this year
by Frobenius in Crelle's Journ., lxxix. pp. 185-247.