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

38 HISTORY OF THE THEORY OF DETERMINANTS

dont la consideration IIous sera utile plus loin. Soit A' le determinant
de ce nouveau syste'me: d'aprebs les n0', (3) et (4), nous aurons
Al =det. (Al, - A2,y A3y.. _A,
~d6t. (Al1, A2, A13, A - A,,
+ det. (Al, A, 2A A A Aj')
On sait que si l'on change les signes des termes d'une colonne
horizontale, le determinant change de signe; done
A' = ( l'-1 det. (A1, A2,. A,, + ( Y-2 l)-det. (A2, A1, A3,. A,,)
+ (- l) n-3 dt. (A3, A1,y A 2, A4...A,,)+.........
+ (- l) d t. (A,,1, 7 l,A2..., A,,.2, A,,) + da't. (A,,, All A2, y A,,.1).
Dans la premiere parenthebse, il n'y a pas d'inversion; dans la seconde,
ily a une inversion, etc.; done
The theorem thus reached may be enunciated as follows:-If
fro~m a determined A of the n" order, 'we formn another A' -such
that the first row of A' is the sumn of all the rows of A and every
other row of A' is got by subtracting -the corresponding row
-of A from the row _preceding it in A, them'?
-in Catalan's notation it is
= ( 4l)-'nl. dt. (A1, A 2,..., IAn),
-although, strange to say, it is never so formulated by hini.
A generalisation of it is next given by saying:
"Si la premie're ligne du systo'me (B) avait renf erme' seulemenit p des
qualitites A1, A.2 A,,, nous aurions trouve, pour la determinant
de cc syste'me,
and then there follow a number of applications to the evaluation
of certain special determinants. Thus, to take the siniplest example, having
A 1... 1