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

452 HISTORY OF THE THEORY OF DETERMINANTS

where the equivalence of (m), and (m). -, makes the determinant
centrosymmetric. Of less interest are the cases (pp. 13-21) in
which the bases increase regularly by 1 save at one place where
the step is from m3r to m+r+s, and the cases (pp. 24-30) in which
the suffixes increase regularly by 1 save at one place where the step
is from p+u to p+u+v. For such cases the modes of treatment
are as far as possible the same as before. Interest, however, is
reawakened when we come to the determinant
where the w +1 bases m, n,..., y, z are any integers whatever.
Taking first the case where p 0, Zeipel removes the factors,
1 1 1
1 ) 2!' 3!'
from the columns, and shows that the resulting determinant is equal
to the difference-product of m, n,..., y, z, thus obtaining.y, z)
(l2n)o(n)1.. (y)W-l(z),, = 1w2... (ce-1)2w1'
a result obtained the same year by Stern when studying alternants.
Then proceeding in the same way, he obtains for
(v),)p),n lp~j (,y)p+ -1(z)p+.w
the expression
(m)p(n__p... (y)p(_)p (n-p)0(n-p). ( -p). (z p).
WAPp~i + I)P. (P + CV)
and thence, with the help of the previous case,
(,Yn2pn)p... (y)p(z)p (Inn., y, z)
The third Section (pp. 41-57) concerns the principal minors of
arrays like
(m+d)p (m+d)p+l (m+d).~2 (m+d)P+3
(m+2d), (m+2d),~1 (m+2d),+, (m +2d)~3
M.P *, (m+c 1 (M)P2d +2
(m+d), (m+d)p~l (m+d)p~2
(m+2d)p (m+2d),~1 (m+2d)p+2
(m+3d)p (m+3d)p~l (m+3d)p+2