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

174 HISTORY OF THE THEORY OF DETERMINANTS

a, ap, a,..., a, which originate by permutation from a single
product of integral powers of the said variables. The determination of such functions is thus reduced to the problem of transforming T so as to have no longer occurring therein the single
elements a, al, 2.., an, but instead those combinatory sums
of them which are the coefficients of the powers of z in the
development of (z-a)(z-a1)(z-a2).. (z-an) or f(z) say.
Without further preparatory statement the announcement is
made that the solution is readily reached when the relation of T
to the determinants
~~-.~. ~- or A,
(t a)2 (t-a - a)2
is known, namely, the relation
D = T.A.
In proof of this relation it is pointed out that
{f(t) -(tl) (tt)... f(tn)}2 D
being an integral alternating function both with respect to the
elements t, t1, t2,..., t, and with respect to the elements
a, a1, a2,..., a, is exactly divisible by the two differenceproducts
II(t, t&, t2.. tn, H(a, a, aC..., an),
and that although we cannot with equal promptness tell the
remaining factor, we are able to determine it from knowing a
sufficient number of its special values, namely, those values got
by putting each t equal to one of the a's. Since the number
of ways in which the n+1 a's can be taken when repetitions
are allowed is (n+l)n+l, this gives us (n+l)1f+1 values, of
which, however, only two are different, namely, the value
(-1l)2('+l).f(a) f '(al) /'(a2)... f'(aL) obtained in the n! cases
where all the a's used are different, and the value 0 obtained in
every other case. The determination, we are told, can be made
by using an extension of Lagrange's interpolation-formula, the
outcome of the work being