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

62 HISTORY OF THE THEORY OF DETERMINANTS
is even or odd according as
1+"2+. +. ' +aI+31+ 3... +/3n
is even or odd. The application which he afterwards makes of this
in connection with Laplace's expansion-theorem will be readily
guessed.
LAGUERRE, E. (1876).
[Sur la m6thode de Monge pour l'integration des equations lineaires
aux differences partielles du second ordre. Nouv. Annales
de Math, (2) xv. pp. 49-58.]
The first section of the paper (pp. 49-53) is occupied with a pure
problem of determinants, namely, the finding of values for
a, b, c, d, a, 13, y, - so that
1. r s. 1 s t 8
a b c d
may represent
Hr + 2Ks + Lt - M + N (t - s2).
The problem is solved, and the relation between two solutions
discussed.
FROBENIUS, G. (1876).
[Ueber das Pfaffsche Problem. Crelle's Journ., lxxxii. pp. 230-315.]
Frobenius having to deal with the vanishing of the minors of a
skew determinant takes a step backward (~ 4, pp. 239-241) to the
consideration of determinants in general.
He first gives an interesting proof of the fundamental theorem
regarding the evanescence of an array. For better means of comparison between it and Dodgson's proof, let us apply it to the same
simple case, namely, where the array to be proved evanescent is
aL bl c1 dl el
a2 b2 C2 d2 e2
a[3 b3 c3 d3 e3