1899-1900.] Dr Muir on the Theory of Skew Determinants. 215 
sign-factor of (2,3, . . . ,r- 1, r+ 1, . . . ,n) is ( - l) r . By 
hypothesis, he says, the left-hand side 
= ( - l) r (2,3, . . .,r - l,r + 1, . . .,»)•( - l) s (2,3, . . ., * - 1, * + 1, . . n), 
= (- l) r+s (2,3, ..., r- 1, r+1 , ...,«) (2,3, . . ., s-1, s+1, . . 
and therefore by a previous theorem 
= - ( - l) r+s (2,3, . . . , r - 1, r + 1, . . .,%) (3, . . s - 1, s + 1, . . .,rc,2), 
the first term of which is 
( 1) 7 + s «23 • • • ®»-i,n*®34 • • • Qn y 2i 
Or - ( - l) r + s « 2 3 «34 • • • «n-l,^n2 j 
and the right-hand side 
=- cofactor of a rs in 
0*22 «23 • 
• • &2m 
«32 «33 . 
. • Cts n 
^w2 ®w3 • 
• • 
5 
= ( - l) r + s 
a 22 
«23 • • • 
• a 2)S - i 
«2,s+l • • 
• • CL2n 
&32 
«33 • • • 
• ^3,s- 1 
«3,s+l 
• • 0>3n 
a r _i ( 2 
Ur- 1,3 • • • 
1 ^r-l,s+l ' • 
. . CLr-l,n 
«r+l,2 
flr+1,3 • • • 
£*r+l,$+l • • 
. . dr-\-\ y n 
&n,2 
tt/i,3 • • • 
• ^n,s-l 
Ctn,s + 1 • • 
and therefore, on account of the translation of the first column to 
the last place, 
= - ( - 1 y+ s 
«23 
. . . « 2 ,s-l 
« 2 ,s+l 
. . . «2,7l 
«22 
«33 
. . . «3,s-l 
«3,s+l 
• • • <^3,n 
«32 
Or- 1,3 • 
• • • «r-l,s-l 
«r-l,s+l < 
• • • Ofr- 1 ,% 
«r-l ,2 
CLr+ 1,3 . 
«r+l, 8 +l ■ 
• ■ • Mr+l,n 
«r +,2 
«w,3 
• • • Ctn,s - 1 
$n,s+l 
. • • Mn,n 
Ct>n,2 
the first term 
of which is 
-(- 
l) r + s «23 «34 
. . . dn - i,n 
®n , 2 5 
exactly as before. 
