1888 - 89 .] Dr T. Muir on the Theory of Determinants. 
755 
In the paragraph following the above Cauchy proceeds, as it 
were, to rectify matters. He says (p. 162) : — 
“ Les formes des fonctions designees par 
$( x )> x( x \ etc - 
etant arbitrages, aussi bien que les variables 
x,y,z,. . . , 
permettent aux divers termes qui composent le tableau (2) 
d’acquerir des valeurs quelconques, et repr^sentons ces vari- 
ables a 1’aide de lettres diverses 
x f y,z, .... ,t 
affect4s d’indices diffdrents 
0, 1, 2, . . . , ra-1, 
dans les diverses lignes verticales. Alors, au lieu du tableau (2), 
on obtiendra le suivant 
if* w nt* o/* 
'*'2’ * ‘ '• • 3 1 
Vm Vv Vv • • • * > Vn-i 
(5) 1 4 • • • • r 4-1 
A> ° ' 5 4-i 
et la resultante s des termes dans ce dernier tableau sera 
^ = S[ + ^o2/i4* • • 4-i] • 
The general determinant is doubtless here reached, but the transition 
requisite for the attainment of it, viz., from x( x )> . 
to the perfectly independent x 0 , x v x 2 , .... is not made without 
considerable strain. This is all the more surprising, too, when we 
consider, that a much less troublesome and less objectionable mode 
of bringing determinants under alternating aggregates lay ready to 
Cauchy’s hand. Bearing in mind the definition given above, of 
fonctions alternees par rapport a diverses suites , we see that a 
determinant of the % th order could have been made to appear as an 
alternating function with respect to n ranks of n variables each. 
For example, the determinant 
