Dr T. Muir on the Theory of Determinants. 491 
first, in columns, for all the a’s, and secondly, in rows for all 
the a’s. 
The last is 
^ ~ ^rhv-f\ 
or 2 [S”(ap^, J (xliv. 2) 
where is the system adjugate to It is obtained from 
the n.n equations (xliv.) just as they were obtained from the 
n.n equations (33), use being made of the theorem 
M„ = DX 
In concluding, Cauchy refers to Binet’s researches on similar 
matters. Most of what he says in regard to them has already been 
given (see p. 498, Vol. XI Y. above). The rest of it is as follows 
(p. Ill):- 
“ II [Binet] me dit en outre qu’il avait generalise le 
theor^me dont il s’agit [M,, = D„8J, en substituant au produit 
de deux resultantes des sommes de produits de meme espece. 
J’avais des-lors deja d^montre le theoreme suivant : 
D \n systeme quelconque ^equations symetriques on pent 
deduire cinq autres systemes du mhne ordre ; mais on n'e 7 i 
sam^ait dMuwe un plus grand nomhre. 
J’ai demontre depuis a I’aide des methodes precedentes cet 
autre theoreme : 
D \n systeme quelconque d' equations symetriques de V ordre n, 
on pent toujours deduire deux systemes equations symUinques 
de V ordre 
n{n - 1 ) 
2 ’ 
deux systemes d equations symetriques de V ordre 
»(»-!)(« - 2 ) 
1.2.3 ’ 
En ajoutant entre elles les equations symetriques comprises 
dans un meme systeme, on obtient, comme on I’a vu, les 
formules (50), (51) et (70) qui me paraissent devoir etre 
semblables a celles dont M. Binet m’a parle.” 
The last sentence here raises an important question for the 
historian to settle, viz., whether Cauchy is to share with Binet the 
