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

134 HISTORY OF THE THEORY OF DETERMINANTS

when m < n, and thus obtains from the multiplication-theorem
of determinants the result
FI F(x,).- F2(X2)... Fm,,(xm)
= r I ~t(a;) F2(a2)... Fm(am) I i-a)'(x2-a.( )1
0`(aji)~ 95(o(aO ~ ~ (P'(am)
It only remains then to employ on the right the already known
equalities
2 0 M-1 2.I, ~ L, I+~ ~lz
f'(a1) c'M(a2)O. 4'(acm) =a(- 1..m(ma1)a am; am+, a cn)
and
I (- a1) (2- a2)'. (Xm am)' ( n - Pt -1) RX01.2... X: 1 2.a
and on the left the self-evident identity
((X-1). (x2). ~ (X) (Xm) = R(x,,..., I m a,, a2, aj., a~1)
= R(x1,.., X; a1,...,am). R(x.,..., Xm; a,),+,. an)
A remark added by Borchardt, and specially interesting from his
standpoint, is to the effect that when m is taken equal to 1 the
theorem degenerates into
F, (X.) F~al)R(Xj; aZ, a3g.., an)
R(a1; a2, a3,., an)
tbhat is, into
FF(x) FI(al)
R (x.; al, U2, - ") (:L; a2, I, an) - (Xj. ll
which is exactly the simple interpolation-theorem used in the
course of the demonstration.
On our side, however, it must be noted that Borchardt's theorem
is of no practical use if we are seeking for the result of the 'performance of the actual division indicated in the left-hand member,
as the equivalent offered by the other member involves not only
the legitimate x's, but also the n extraneous quantities a1, a2, 2, a,.
For example, when m = 2, n = 3, FI(z) = bz, F2(Z) = cZ2, we
obtain for
bx1 CX,2
bx2 cx22 + (X2-X),