1898 - 99 .] Dr Muir on a Single Term of a Determinant. 443 
Another little step and he might have laid down explicitly the 
rule : — 
If v be the number of interchanges necessary to transform a given 
permutation into the standard permutation (i.e., the permutation 
in ivhich the elements occur in their natural order) the sign of the 
given permutation is ( - ) u . 
This step he did not take, although he begins another corollary 
with the words “ da die Permutation 1, 2, 3, . . . , r allemal das 
Zeichen + hat.” Notwithstanding the omission, however, we shall 
be justified in associating the rule just formulated with Rothe’s 
name. 
(3) The third rule appeared twelve years after Rothe’s. It is 
due to Cauchy, and was published in his great “ Memoire sur les 
fonctions qui ne peuvent obtenir que deux valeurs egales et de 
signes contraires par suite des transpositions operees entre les 
variables qu’elles renferment.” * 
We are now required to count the number of circular substitu- 
tions which are necessary for the transformation of the given 
permutation into the standard permutation. Cauchy’s words are 
(P- 56)- 
“Soit 
a *\ a P2 a £n 
le produit symetrique dont il s’agit, et designons par g le 
nombre des substitutions circulaires equivalentes a la sub- 
stitution 
/I 2 3 
v* P 7 V- 
Ce produit devra etre affecte du signe + , si n - g est un 
nombre pair, et du signe - dans le cas contraire.” 
For example, if the permutation whose sign is wanted be 
68319254 7, we write above f it the standard permutation 
123456789, thus obtaining the substitution 
7>1 2345678 9\ 
\6 8.3 1 9 2 5 4 7/; 
* Journ. de VEc. Potyt., x. (Cah. xvii.) pp. 29-112. 
t Or below it. The arrow-head is useful in this connection. 
