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

456 HISTORY OF THE THEORY OF DETERMINANTS

and then showing that the determinant as thus reduced is equal to
r!. Attention is of course drawn to the fact that the r quantities
a, /3,.., p introduced through the distributed multiplication do
not appear in the expression of the final value.
The main result of the next two Sections (pp. 9-17, 18-24) is
that If each element of the determinant
(m)k (m)p1 (m)p+l.... ( m)p+
(m+l)k (m+l)p (m+1)+l.... (m1+l)p+r-1
(m+r)k (m- +r)p (m-+r)p. (m+r).+r_,
where k is equal to one of the other suffixes, be multiplied by the corresponding element of the array
1a 3R.... p
1 a+l P3+l.... p+1
1 a+r,3+r.... p+r,
the resulting determinant is equal to
(k-p +l)(k-p+2). r. (m +1.. (m) +r)- 71),(m +r)1,
* (m-p —a)(m-p-/3-1)(mn-p-y-2)...,
where the number of factors following the fraction is k-p. The proof
is accomplished in instalments, the case where p = 0 being taken
first and being made dependent on the case where p = 0 and r = k.
The mode of showing that in this latter case the value is
(m-a)(m-/3-l)(m —y-2)...
is to prove that the determinant then vanishes if m be put equal
to a, /3+1, y+2,...; some shorter mode ought to be devisable.
The other result worthy of note in the third Section is that If each
element of the determinant
1 (m)o (m).... (m)_
1 (m+d)o +d)d)..... (m+d). r
1 (m+rd) (m+rd.. (m+rd)r_ (m+rd