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

466 HISTORY OF THE THEORY OF DETERMINANTS

CUNNINGHAM, A. (1874).
[An investigation of the number of constituents.. Quart,
Journ. of Sci., (2), iv. pp. 212-228.]
In the sections (III.-VI.) which concern zero-axial determinants
the fresh subject investigated is the number of terms in an n-line
determinant having r zeros in the diagonal. One expression obtained
for this is,(n,) + (n-'i)1(r -(n- ) + (n-')2(n — 2) +.
as we should expect; but by a process of' symbolic inversion' he
derives another therefrom, namely,
n' - (r)1.(-1)! + (r)2.(a-2) -.
which of course includes the familiar
n!.n!
V (') 2= i! +
There is also obtained a recurrence-formula, which we may write
in the form,r = -n,r+1 -+ [ n-1,r
As an alternative source for the second expression for u,,, there is
given a very interesting expansion for a determinant having r zeros
in the diagonal, namely,
A ( ax ) + a( Dxayy Vxxa )
- t('LrJ"'wZZa xxDa aa
+ ( 1 ) z (ai..2.... a. a...c... *
where A is a1a22... a.. Although the truth of this is said to
be easily seen, it may be well to note that the zero elements are
taken to be in the first r places of the diagonal; that x, y, z,... are
not greater than r; that the 2 in the last term is not required; and
that as an example we have