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

444 HISTORY OF THE TIEORY OF DETERMINANTS
where h, Jc, m, n are in ascending order of magnitude, the author
eliminates K(h,k-l1) and obtains
K(h,mn) K(k,m ),K(h,1O-2) K(k,in) K(kc+l,nm)
= a,_,,K<(7Ll-2). (a)
K(h,n) K(kc,n) ' K(kc,n) K(kc+l,n)
Then by taking the particular case of this where 7c appears in
place of h and k +1 in place of kc there results
K(]~,m) K(k+lm) K(k+l,m) =(] +2,m)
K(7c,n) K(k+l,n) K(+ln) K(+2)
which when applied to one of the determinants occurring in itself
gives
K(c,nm) K(l +l,m) K(lk+2,m) K(lc+3,j)
K(k,n) K(k +1,n1) +a k+1,2 K(l +2,n) K(lc+3, n)
and finally
K(m+ 1,n) K(rn+2,mn)
a77l ak+,E+2,1 K(m+Il, ) K(mn+ 2,n)
= Cak,k+C k+1,7,+2....,,+ K(n- + 2,n). (3)
Further, by using this to make a substitution in the previous
result (a) there is obtained
K(h,?r) K(k,nm), = a-1,k_,k+1.... aI,,+. K(h,Jc- 2)K(m+ 2,a), (y)
which on putting 7 = h + 1 and m = n -1 becomes
K(h,a-1) K(h+1,n-1)
T^7 \ = h,7+-1h+-1,h-2.... -- 1,n
K(h,?) K((h + 1,n) = l 2...
-a result which may be compared with one of Schlafli's and
Worpitzky's, but which is more general in that the main diagonal
of each " K-Determinant" does not consist of units.