202 Proceedings of Royal Society of Edinburgh. [skss. 
regarding a bordered skew symmetrical determinant of even order. 
It will be remembered, however, that in the statement of this 
latter theorem, the peculiar narrow use of the word £ borde ’ did 
not occur. 
Although what may be called Part Second of the paper (pp. 
301, 302) may seem at first sight to concern something else, it 
really only draws attention to the fact that the minors (by which 
he means those afterwards named primary minors) of a shew deter- 
minant are themselves shew , being u gaudies or din air es ” when their 
cofactor in the original determinant is of the form rr, and “ gaudies 
hordes ” when their cofactor is of the form rs. Considerable space 
is occupied in verifying by two examples that the same result will 
be reached whether we apply the theorem of Part First directly to 
123 .. . n | 123 . . . n 
or to the primary minors in its equivalent 
IP 23 ... n | 23 ... n - 12- 23 ... n | 13 ... n + 
What may be called Part Third (pp. 303-305) is very forbid- 
ding, by reason of the defective mode of exposition and of the 
awkwardness of the notation employed. Probably this accounts 
for the fact that the interesting theorem which it contains has 
never emerged until now from its place of sepulture. A portion 
of it must of necessity be given verbatim, if only for the purpose 
of preserving historical colour. It commences — 
“ Je remarque que le nombre des termes du developpement 
(p. 299) du determinant gauche est toujours une puissance 
de 2, et que de plus, ce nombre se reduit a la moitie, en 
reduisant a zero un terme quelconque aa. Mais outre cela, 
le determinant prend dans cette supposition la forme de 
determinant [gauche] dun ordre inferieur de l’unite. Je 
considere par example le determinant gauche 123 | 123. 
En y faisant 33 = 0 et en accentuant , pour y mettre plus 
de clarte, tous les symboles, on trouve 
123 | 123' = ll'.(23') 2 -|-22'-(13') 2 . 
