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

DETERMINANTS IN GENERAL (JANNI, 1876, 1879) 61

JANNI, V. (1876, 1879).
[LEZIONI DI ALGEBRA COMPLEMENTARE: Teoria dei Determinanti.
(v)+50 pp. Napoli.]*
[LEZIONI DI ALGEBRA COMPLEMENTARE. (viii)+228 pp. Napoli.]*
For its extent (50 pp.) this is an unusually pleasing exposition.
Most of the points worth noting, however, have already been
attended to in dealing with the same author's paper of 1874.
Trudi (1862) is followed in making the sign of a term of a determinant dependent on the combined number of inverted-pairs in
the two sets of suffixes; but Janni shows in addition that the interchange of two double-suffixed elements will make no alteration in
the sign as thus determined; that therefore the order in which the
elements may happen to be taken in the formation of the term is
of no real consequence; and likewise that this order may be made
such as to necessitate the counting of inverted-pairs in only one
of the two sets of suffixes.
Trudi's theorem regarding a partitioned permutation is established only for the case in which the permutation
alag2....,, I a +I... a
has no inversions in either of its two parts taken separately, and
where, therefore, the total number of inverted-pairs is
al+a2+... +. am-m(mn+ 1);
but Janni proceeds to take another permutation
1P312... 1 | /3Mn+l1... do
similarly partitioned and similarly in part arranged, and, asserting
that the number of its inverted-pairs is
/31+2 +.. + /3 ( + m — (+),
is thus able, by reason of m (m+1) being even, to conclude that the
combined number of inverted-pairs in the two permutations
ala2.... an J an * * *.aI
/3132. '. (m | 4/+n1. '. *
* The first of these books seems a sort of trial edition of cap. i., ii., iii. (pp. 1-64)
of the second; but in neither is reference made to the other, and the publishers
are different.