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

ALTERNANTS (BINET, 1837)

1577

f(x) + F(x) can be partitioned into n fractions having x - A, x - B,
x-C...for denominators, the result as given by Euler being.f(x) _ f(A) f(B) f(C)
flx) + i~
F(x) - (x-A)f'(A) (x - B)f'(B) (x - C)f'(C)
Substituting successively K, K1, K2I...for x in this, a set of
equations is obtained from which it is seen that the solution
of the set
a b c f(K)
-- -+ + + 5.
K-A K-B K-C F(K**
a b c f(K1)
K, -A+K-B+ - +.. F(K(,)'
a b c f(K2)
K2 -A K2 - K22
a ()f(A)
F F'A) (A(-B)(A-C)(A-D)...'
b (B) f(B)
F'(B)- (B-A)(B-C)(B-D)....
Now the set of equations here solved is more general than that
with which we started, the latter being the particular case of the
former where
f(K) f(K1) 1.
F(K) F(K1)
To effect this specialisation it is only necessary to make the
arbitrary function f (x) equal to
(x-A)(x-B)(x-C).... - (x-K~)(x-IK1)(x-K2)....
or equal to
F(x)- f(x) say;
(where, be it observed, the condition as to the degree of f(x) is
fulfilled); for then, since f (x) vanishes when x K, K1, K2
we have
f(K) = F (K), f(KI) = F (K1), f(K2)= F(K2).
The corresponding change in the values of the unknowns is easily