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

76 HISTORY OF THE THEORY OF DETERMINANTS
vv v
VASCENKO-ZACHARCENKO, M. E. (1878).
[THEORY OF DETERMINANTS AND THEORY OF FORMS. (In Russian.)
vi+501 pp. Kieff.]
Ostensibly there is here devoted to the theory of determinants
240 pages, but much of the space is occupied with algebraical matter
that is not closely relevant. On the main subject there is nothing
calling for attention. The tables of symmetric functions and
eliminants at the end (pp. 220-240) are apparently reproduced from
Salmon's Modern Higher Algebra.
WOLSTENHOLME, J. (1878).
[MATHEMATICAL PROBLEMS,... (pp. 275-280). 2nd edition.
x+480 pp. London.]
Wolstenholme * gives the 5-by-6 array
j... a b c
z ab
I.. c
\X... b c ~
x y z.,
which conveniently illustrates Dodgson's test for 'evanescence'
(see above, p. 25). The first two primary minors can be shown
to vanish by diminishing in each case the last row by the sum of
the three immediately preceding rows, and they are easily seen to
have a common 4-line minor which is not equal to 0.
FALK, Mi. (1878).
[Sur une propri6te des determinants nuls. Nouv. Corresp. Math.,
iv. pp. 373-376.]
The property in question is that if a determinant be equal to 0,
it is possible to transform it so as to have a row or column of zeros.
By way of proof it is pointed out that the determinant being a an
the number of sets of non-zero values of x,, x2.,, x which will
make the expressions
a61lll+aC-12x2- ~ ~ +a~ -
a2lX +a22X2 +. * * * ~2nXn,
a,.xl +am2X2 +... +a.nnXn,
i- But it appears in Cambridgye Univ. Exam. Papers, iv. (1875), p. 224.