The theory of determinants in the historical order of development, by Sir Thomas Muir.

284 HISTORY OF THE THEORY OF DETERMINANTS
and therefore by a previous theorem
= -(-l)r+ s(2,,...,,-,,+1...., )(3s,..,s-, s+,...,2, 2),
the first term of which is
- (- l)r+S. ~a,-1l23. 3... a, * 2
or - ( — l).+a2334.a n-1,_ln2;
and the right-hand side
= cofactor of ar in a22 a23 a2n
a32 a33... a3nt
an2 ~a'na3 ~* * * (nn
= ( —)r+s a22 a23. *.. a 2,s- a2,s+1,... a2
a32 a33.... a3,-l a3, s+.... a3,
a -1,2 a..-1,3... a ls-l a, l,s+....
Cr+1, 2 a(,+1,3..(-+, s -1 a-+l, s+l}- ar+l,
an,2 an, 3..,* s-1 aw,s+1. * n,n
and therefore, on account of the translation of the first column
to the last place,
=- (- 1)"+s a23.... a2,,- a2,8s+1.... a2,n a22
a33.... a3,s-1 a3,s+l1.... a3,n a32
ar-1,3... ar' -ls-l (,r-l,s+l ar -l,n ar-1,2.r+l13. *. * -. a+l,s':-1 -+1,8+1.... ar+l,n at+l,,2
an,3.... ats-l an, s+l. *. an,n a,2
the first term of which is. - -(-l1)r+sCa23 634... Cl,,_l.,aL,2,
exactly as before.
To the proof no note is appended drawing attention to the fact
that the very same result would have been reached by taking