1898-99.] Dr Muir on a Single Term of a Determinant. 441 
Determination of the Sign of a Single Term of a 
Determinant. By Thomas Muir, LL.D. 
(Read January 9, 1899.) 
(1) As is well known, the first rule given for ascertaining the 
sign of a single term of a determinant was made known by Cramer 
in his Introduction d V Analyse des Lignes Courbes algebriques , 
published at Geneva in 1750. On page 658 he says — 
“ On donne a ces termes les signes + ou - , selon la Regie 
suivante. Quand un exposant est suivi dans le meme terme, 
m4diatement ou immediatement, d’un exposant plus petit que 
lui, j’apellerai cela un derangement. Qu’on compte, pour 
chaque terme, le nombre des derangements : s’il est pair ou 
nul, le terme aura le signe + ; s’il est impair, le terme aura le 
signe - . Par ex. dans le terme Z 1 Y 2 Y 3 il n’y a aucun 
derangement : ce terme aura done le signe + . Le terme 
Z 3 Y 1 X 2 a aussi le signe + , parce qu’il a deux derangements, 
3 avant 1 & 3 avant 2. Mais le terme Z 3 Y 2 X 1 qui a 
trois derangements, 3 avant 2, 3 avant 1, & 2 avant 1, aura 
le signe — .” 
According to Cramer, therefore, If 8 be the number of derange- 
ments in the 'permutation corresponding to any term, the sign of 
the term is ( - ) d . 
Instead of the word “derangement,” Gergonne * in 1813 used 
“ inversion ” in speaking of Cramer’s rule : Cauchy f did the same 
in 1841: and, consequently, the latter term or “inversion of 
order” has come into pretty general use. “Inverted-pair” is 
probably a still better expression. £ 
(2) The next rule originated with Rothe, having been fore- 
shadowed by him in the second volume of Hindenburg’s Sammlung 
combinatorisch-analytischer Abhandlungen , published at Leipzig in 
* Annates de Math., iv. pp. 148-155. 
t Exercices d analyse et de phys. math., ii. pp. 145-150. 
t Proc. Roy. Soc. Edin., xvi. p. 449. 
VOL. XXII. 7/4/99 2 F 
