Dr T. Muir on the Theory of Determinants. 
483 
Similarly 
(4^5o) 
stands for the system of which the determinant is 
2^3-31 J i^l’3^2’4^^'3’5l 
• • ? 1 *^ 2 - 3 % 4^*5! 
I^2'1^4*2^5*3' » ’ 1^2* 2^4* 4^5’ 5! » I*^2‘3^4‘4^5'5l 
1^3*] '^4* 2^5' 3! ’ ’ l^3‘2^4’4^5‘5l ’ k3'3^4'4^5‘5i 
and which is called the “ complementary derived system.” (xll 3.) 
To every “ terme ” of the latter there corresponds a “ terme ” of 
the former, the one “terme” consisting exactly of those a’s of the 
original determinant which are awanting in the other. This relation- 
ship Cauchy goes on to mark by means of a name and a notation. 
He calls two such “termes,” and for example, 
“ termes complementaires des deux systemes ; ” (xli. 4) 
and if the symbol for the one be by previous agreement 
jU.TT 
the symbol for the other is made * 
(XLI. 5.) 
As for the signs of the “ termes ” in “ derived systems,” Cauchy’s 
words are (p. 98) — 
“ En general, il est facile de voir que le produit de deux 
termes complementaires pris a volonte est toujours, au signe 
pres, une portion de ce meme determinant (D,j). Cela pose, 
etant donne le signe de I’un de ces deux termes, on deter- 
minera celui de I’autre par la condition que leur produit soit 
affects du meme signe que la portion correspondante du 
determinant D,^.” 
All these preliminaries having been settled, the weighty matters 
of the section are entered on. The first of these is a complete and 
* If Cauchy had adopted a slightly different principle for determining the 
order of combinations, the 114*^ combination of ^ things and the (P-ac, + 1)**^ 
combination of n-p things would have been mutually exclusive, and the con- 
vention here made in regard to notation would have been unnecessary. 
