1887.] Dr T. Muir on the Theory of Determinants. 
509 
termes seront done respectivement egaux a ceux qu’on obtient 
en developpant le determinant 
D w 8>( db ^2 - 2 • • • • a n . ri) ) 
et comme le produit principal a vl a 2 . 2 . . . a n . n est positif de 
part et d’autre, on aura necessairement 
D w = S[ =•= a n . n S( =*= dyi a 2 . 2 .... a n _ l . n _ 1 )] (vi. 3) 
d'n’ifn’n d“ ^■n-\“nhn-\'n d" • « • • d“ « 
En general, si Ton designe par /x Tun des indices 1, 2, 3, . . . , n 
on trouvera de la meme maniere 
D w = S[ =*= ^ ^1*1 ^2’2 • • • • ^fJL- l^/u.+1'M+l ' ' ' 
(VI. 4). 
..... Cette derniere equation 
0 = a i'J>i > d- d- . . . d- a wv b n> (XII. 6) 
sera satisfacte toutes les fois que v et /x seront deux nombres 
differens l’un de l’autre. 
.... on aura done aussi 
D „ = a /t . 1 6 j x. 1 d- a M . 2 & M . 2 + . . . . d- (vi. 4) 
0 = a v .jb^ + a v . 2 b fJi . 2 + . . . . -i- a v . n b fJL . n (xii. 6) 
les indices [x et v etant censes inegaux.” 
The expressions here denoted by b vl , b v2 , . . . . are spoken of 
as adjugate (“ adjointes”) to a vv a v2 , . 
. . ; and the system 
f ^1*1 \'2 
J ^2-1 ^2-2 
j &C 
. b 2 . n 
5 n . j b n . 2 
• i*n'n 
as adjugate to the system ( a Vn ). Similarly the system {b n .j) is said 
to be adjugate to the system (a n .j) ; and, on the other hand, it is 
said to be adjugate and conjugate to the system (a Vn ). 
Up to this point no new property has been brought forward. 
The following paragraph (p. 68), however, opens new ground, the 
formula given in it being of some considerable importance in the 
after development of the theory. 
“ Si dans le systeme de quantites (a^j) on supprime la derniere 
