506 
Proceedings of Royal Society of Edinburgh. [july 18, 
f a v\ 
a 2 'l 
a 2,'\ • • • • 
j a V 2 
(^2* o 
a 3 . 2 • • • • 
i a vz 
CO 
03 
(X3.3 . . . . 
j &C. 
1 a V n 
a 2' n 
a^. n . . . . 
Cf*n % \ 
^k'3 
C^n'n 
derived from the previous system by interchanging the suffixes of 
each “ terme ” is said to be conjugate to the previous system. A 
symbol for each of these systems is got by taking the last “ terme ” 
of its first “ suite horizontal, ” and enclosing the “ terme ” in 
brackets : in this way we are enabled to say that {a v j) and (a n .j) 
are conjugate systems. 
In the course of these explanations a modification of the rule of 
term-formation is incidentally noted, the form taken being specially 
applicable when the quantities of the system have been disposed in 
a square. Cauchy’s wording of this now familiar rule is (p. 55) — 
. .... “ pour former chacun des termes dont il s’agit, il 
suffira de multiplier entre elles n quantites differentes prises 
respectivement dans les differentes colonnes verticales clu 
systeme, et situees en merne temps dans les diverses lignes 
horizontales de ce systffine.” 
Here we may note in passing that the disposal of the “ termes ” in 
a square might have enabled Cauchy to point out (which he did not 
do) the difference between Gauss’ use of the word “ determinant ” 
and his own, by saying that the “ determinant of a form ” had its 
conjugate u termes ” equal. 
The rule of signs applicable to alternating functions in general 
is modified for the special case of determinants, and takes the 
following form (p. 56) : — 
“ l^tant donne un produit symetrique quelconque, pour ob- 
tenir le signe dont il est affecte dans le determinant 
S( i ^Tl ^2' 2^3*3 • • • • ^ n'n ) 
il suffira d’appliquer la regie qui sert a determiner le signe d’un 
terme pris a volonte dans une fonction symetrique alternee. 
Soit 
a>a ci 1 3.0 a^. n 
le produit symetrique dont il s’agit, et designons par g le 
