Dr T. Muir on the Theory of Determinants. 
505 
From a modem point of view there are but two which are really 
different, viz., 
\ah'\.\ac' d"\ - \ae'\.\ah'd"\ + \ad'\.\ah'c"\ = 0 
and a\hcd”\ - h\acd''\ + c\ah'd"\ - d\ah'c!'\ — 0 , 
the twelve quantities concerned being 
abed 
a' h' c' d' 
a" h” c" d" . 
The former is obtainable from B^zout’s identity 
\aVc"\.\de'f'\ - \ab'd"\.\ce'f'\ + \ac'd"\.\heT\ - \hdd"\.\ae'f"\ -= 0 
by putting 
==0,0, 1 
and e , e, e" = a , a', a”. 
The other, as is well known, comes from Vandermonde. 
Before proceeding to the case of four unknowns, a notation is 
introduced in the following words (p. 6) : — 
“ Soient a, b, c, d, f, g, h, etc. des lettres representant des 
quantit4s quelconques ; k, I, m, p, q, r, etc. des indices d’accens 
qui doivent etre places a la droite des lettres. Au lieu de 
mettre ces indices comme des exposans, plagons-les au-dessus 
des lettres qu’ils doivent affecter, de manike qae designe a 
affecte du nombre k d’accens ; que ^ ^ indique le produit 
k I • k I k I 
de ^ par ^ ; ainsi de suite. Eepresentons la quantite Jj- ha 
par de sorte que nous ayons cette equation 
f k l\_k l k i ” 
\abj~ ab ~ b a . 
This being settled, the similar quantities of higher orders are 
defined by the equations 
/k l 77i\ mfk l\ l/k m\ kf I m\ 
yab c\abj ~ c\a b j c\a b j ^ 
I k l mp\ pf k I m\ mf k I p\ l/k mp\ k / I m p\ 
\ab cdj~ dya b cj ~ d\a b cj d\a be) ~ d\a b cj ^ 
tfec. &c. tfcc. 
It is thus seen that Desnanot’s definition is exactly the same as 
