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

120 HISTORY OF THE THEORY OF DETERMINANTS

CATALAN, E. (1878).
[Theoremes de MM. Smith et Mansion. Nouv. Corresp. Math.,
iv. pp. 103-112.]
This is a fresh and clearly written exposition, the last of Mansion's
theorems, however, being altered for the worse into " Tout produit
est egal a un determinant."
STODOLKIEWICZ, J. (1878).
[Proof of Cayley's formula for calculating the number of different
kinds of terms in an axisymmetric determinant (In Polish).
Izwiestja Cesarskiego Uniwersytetu Warszawskiego, 1878,
No. 6.*]
The formula referred to is the difference-equation
~ = nu_ - - (n-1) (n-2)_ 3,
which is now established by merely using elementary properties
of determinants. Putting v, for the number of different kinds of
terms in the special form of n-line determinant which is only axisymmetric after a row and a column have been deleted,t we see,
from familiar developments of the two determinants, that
U, = u.+_1 +(n —1)un_2 + -(n-1)(n-2)v_-2,}
Vn = un-1+(n-l)v, 1;
and between these the v's can readily be eliminated, with the desired
result.
PAIGE, C. LE (1878).
[Sur un theoreme de M. Mansion. Nouv. Corresp. Math., iv.
pp. 176-178.]
Taking Mansion's determinant for Y1Y2... Ye Le Paige performs
the operations
col-col, col3-col,...., cole-col1;
col4-col, col6 —col2;
col6-col3;
* Or Baraniecki's text-book, chap. x. ~ 94.
t e.g. an axisymmetric determinant unsymmetrically bordered.