1904 - 5 .] 
Dr Muir on General Determinants. 
915 
phraseology, — gives the expression for a determinant in terms of 
its own devertehrated coaxial minors and its primary diagonal 
elements. In the actual wording of the description of the theorem 
it is, not unnaturally, applied to a skew determinant only ; hut 
there is clearly nothing in the nature of the case to confine it to 
this special form. The description is — 
“En effet, soit O le determinant gauche dont il s’agit, cette 
fonction peut etre presentee sous la forme 
O = O 0 + + 0 2 X 22 + • • • +0 12 X n A. 22 + • • • 
oil Q 0 est ce que devient fi si A n , A, 22 , . . . sont reduits a 
zero. Oj est ce que devient le coefficient de A n sous la meme 
condition, et ainsi de suite ; c’est-a-dire : O 0 est le determinant 
forme par les quantites \ rs en supposant que ces quantites 
satisf assent aux conditions (2),* et en donnant a r les valeurs 
1,2,..,%. Oj est le determinant forme pareillement en 
donnant hr, s les valeurs 2 , 3 ,...,%; Q 2 s’obtient en 
donnant a r , s les valeurs 1,3,. . . , n , et ainsi de suite : 
cela est aise de voir si Ton range les quantites X rs en forme 
de carre.” 
Cauchy (1847). 
[Memoire sur les clefs algebriques. Exereices d\ Analyse et de 
Phys. Math., iv. pp. 356-400, §11.] 
In this longish memoir it is the second section (§ ii.) that is of 
interest to us, its title being “ Decomposition des sommes alternees, 
connues sous le nom de resultantes, en facteurs symboliques.” 
The only previous writing with which it is clearly connected 
in subject is Grassmann’s of the year 1844. To ensure the 
possibility of proper comparison between the two, it is necessary 
to do now as was done in the previous case, viz., to give the 
opening paragraph verbatim. No general explanation of c algebraic 
keys ’ need be offered, all that is requisite for our present purpose 
being obtainable from the paragraph itself. It runs as follows 
These are \rs— ~ A srG 0 » 
