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

32 HISTORY OF THE THEORY OF DETERMINANTS

lent of j alb2c3 14. The said determinant he multiplies row-wise
by 2 1 b, 1 * I b2C3 I* | b3c, in the form
1 1....2. 1 c..b
obtaining a determinat eqal o 3
2 I bc | | b31 | | b2c3|
obtaining a determinant equal to
2 I ab23 14 b12 b22 b32
c12 c22 C32
blcl b2c2 b3c3;
and, as the last three-line determinant is equal to
I bc2 * b2c3 1 I b* 11,
the desired object is obtained.
JAROSENKO, S. (1871).
[Teorija opredelitelej i jeja prilozenija. I. Teorija opred'litelej.
xii +159 pp. Odessa.]
This text-book, which has almost the same title as Sperling's of
1858, contains a very full account of the subject, the fullness being
possible by the allocation of so much space to the theory alone. It
opens with a historical introduction of ten pages, and then follow
three chapters (pp. 1-74) on determinants in general and other
three on special forms, that on Compound Determinants (chap. iv.)
being more than ordinarily full.
CALDARERA, F. (1871).
[Nota su talune propriety dei determinanti, in ispecie di quelli a
matrici composte con la serie dei numeri figurati. Giornale
di Mat., ix. pp. 223-232.]
Caldarera's fundamental theorem is Chio's of 1853, regarding
the reduction of the order of a determinant, and is established in