510 Frocecdings of Royal Society of Edinburgh. 
vary. As we should now say, the difference is a mere matter of rows 
and columns. The derived identities (B"), (C")j (B"), . . . are 
consequently found to be quite the same as (B), (C), (D), .... 
The fourth and last source made use of is the well-known theorem 
regarding the aggregate of products whose first factors constitute 
what Cauchy would have called a “suite verticale,” and whose 
second factors are the cofactors, in the determinant of the system, 
of another “ suite verticale.” Desnanot however, viewing the 
theorem from a different stand-point, enunciates it as follows 
(p. 26) 
“ Si Von a n lettres ab .... cdf, et qu'on les combine n — 1 « 
n-1, on aura n arrangemens ab .... cd, ab .... cf, ab .... df, 
a .... cdf, b .... cdf; qiVon aqiplique dans 
cliaque arrangement Zes n - 1 indices kl....mp, ce qui donneva 
ces quantiles 
Jc l . . m ‘p\ / k I . . m p\ / 7c I .. mp\ / k I . . m'p\ /k I ..mp\ 
ab . . c dj ^ yab . . cf) i \^ab . . d f) ya . . c d f) i yb . . c d fj > 
et qu^ensuite on les multiplie cliacune par la lettre qui iVeiitre 
pas dans Varrangement en Vaffectant dVun meme indice et 
donnant au produit le signe plus ou le signe moins, suivant que 
la lettre multiqglicateur occupe un rang impair ou pair dans les 
n lettres^ en partant de la ch'oite, la somme des qrroduits sera 
zero.'' (xii. 9). 
Before proceeding to deduce others from it, he gives a proof of it 
for the case 
The method of proof is interesting, because it depends almost 
entirely on the definition which Desnanot follows Vandermonde in 
using. It will be readily understood by seeing it applied to the 
simple case 
b-f>2^fl^ — bfb-pyd^ -j- bfb-p^d^ — b^ ~ ^ * 
Expanding each of the determinants \b. 2 cy:l^, \b-pfl ^ , in 
terms of the 5’s and their cofactors, we have 
