Dr T. Muir on the Theory of Determinants. 
495 
n(n-\) - 2 ) . . . {n-p + \) 
1 . 2 . 3 . . . • 
Leaving Cauchy, let us now return to Binet, and in order that 
the comparison between the two may he complete, let us formally 
enunciate in all its generality the latter’s theorem also. Binet him- 
self did not do this. After dealing with the case in which the de- 
terminants involved are of the 2 nd order, he merely added (p. 289 ) — 
“ On aura encore pour les integrales 
des resultats semblahles, savoir, 
S(^V,D} 
_ g I zv%x^ + ^z^^xv^y^ 
^ 1 - - %y^'%xv%zl - %z^%yv%x ^ , 
= ^f'%tr%xi'%yv'%z^+ %XT%y^%tv%z^ -f &c.} 
&c.” 
With the help of modern phraseology, the general theorem thus 
intended to be indicated can be made sufficiently clear. Binet in 
effect says : — 
Take s rectangular arrays each with m elements in the row and n 
elements in the column, m being greater than w, viz. — 
(ai)ii(«i)i 2 • 
• • (^l)lTO 
(^1)21(^1)22 
. . . {a^ 2 m • • 
(<^l)sl(<*l)s 2 • • 
■ («l). 
(^2)11(^2)12 • 
• • (^ 2 )lw» 
(<^2)21(^2)22 
. . . {a^ 2 m • • 
(<* 2 )sl(^ 2 )s 2 * • 
• («l), 
(<^«)ii(<^m)i 2 • 
(<^ 9 ^) 2 l (<*«)22 
. . . {a^ 2 m j • • 
(<^w)«i(<^m/s 2 • • 
• («..). 
and other s rectangular arrays of the 
same kind, viz. — 
• • (^l)lw 
(^1)21(^1)22 • 
' • • (^l) 2 >w • • 
(^l)sl(^l)s 2 • * 
• (h). 
(^2)11(^2)12 • 
• • (^ 2 )l»» 
(^2)21(^2)22 • 
' • • {^2)2^ • • 
(^ 2 )«i(^ 2)«2 • • 
■ 
(^^i)ll(^n)l 2 • 
. . (^w)lOT 
(^») 2 i(^m )22 • 
• • (^n) 2 mi 
■ ■ 
■ (&„), 
