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

COMBINATORY NUMBERS (ZEIPEL, 1871)

459

A similar procedure 'suffices to effect the removal of the factor
mr+/3 +2, and thereafter of the others in order, when it is found that
the determinant as thus reduced is equal to 1.
TIRELLI, F. (1875).
[Quistione 36. Giornale di Mat., xiii. pp. 167, 225: solution by
A. Landriani, xiii. pp. 356-358.]
Tirelli's results are somewhat obscured by his notation and by his
adoption of an awkward order for the columns of his determinants.
If instead of using one notation for n(n+l)... (n+k-l)/k! and
another for n(n-1)... (n- k+l)/kl, we use the same for expressing
both, denoting them respectively by
(n+k —1) and (n),
and if we reverse the order of the columns in each case, the identities
submitted by him for proof are:

1 (p+l)l (p+2)2....
I (p+2)i (p+3)2..
1 (p+n), (p+n+1)2....

(p +n -1)_,,
(P+n)n —1 = 1,
(p +2n-2),-_

31 4... Hi 21 31 41... Hi
3 2 42... 2 = 32 42 52... (n.-1)2 = A,..... *An-l (n)n-l (+1),n- (n +2),,_... (2n-2)n_l
With the first determinant we are already familiar from Zeipel.
The third, as we also know from Zeipel, is transformable into the
second by repeatedly diminishing each row in backward order by
the row in front of it, the number of rows affected being one fewer
each time; and the second, by the like operation performed on its
columns, is changed into the persymmetric continuant
2 1...
1 2 1.... 1 2 1....n-l,
the value of which is n.