756 Proceedings of Royal Society of Edinburgh. [sess. 
could have been introduced as a function alternating with respect 
to any two of the three ranks, 
<h a 2 a 3’ 
\ b t , 
^2 ^3 i 
and indeed, as we know, it is alternating also with respect to any 
two of the ranks 
a 1 Zq c 1 , 
^2 ^2 C 2 ’ 
^3 C 3 J 
that is to say, according to another phrase of Cauchy’s, used above, 
it is alternating with respect to the indices 1, 2, 3. 
The fourteen pages (pp. 163-176) which follow, are taken up 
with the properties of determinants as thus defined and with the 
application of them to the solution of simultaneous linear equations. 
Most of the matter is already familiar to us, and may be altogether 
passed over. One of the theorems it is necessary to give verbatim, 
not because of its importance, but because it serves to make evident 
the untenable position Cauchy had taken up in so peculiarly 
bringing determinants under the head of alternating aggregates. 
The theorem is (p. 164) : — 
“Si, avec les variables comprises dans le tableau (5), on 
forme une fonction entiere, du degr£ n, qui offre, dans chaque 
terme, n facteurs dont un seul appartienne a chacune des 
suites horizontales de ce tableau, et qui soit alternee par 
rapport a ces m§mes suites, la fonction entiere dont il s’agit 
devra se reduire, au signe pres, a la resultante s.” 
This not only justifies the definition proposed above to be sub- 
stituted for Cauchy’s, but it also entitles us to say that Cauchy 
having started by including determinants among alternating func- 
tions of one kind, viz., functions alternating with respect to every 
pair of n variables, soon succeeds in showing that they are alter- 
nating functions of an entirely different kind, viz., functions 
alternating with respect to every pair of n ranks of variables. 
The only other noteworthy matter is a theorem in regard to 
