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

158 HISTORY OF THE THEORY OF DETERMINANTS

made: for example, in the case of a, we have only to substitute
F(A)-f(A),-or, what is the same thing, -f(A),-for f(A) in
the numerator, the result being
(A-IK)(A-K,)(A-K2)....
a =- (A-B)(A-C)...
We have thus as Binet's theorem:The solution of the set of equations
+-~~+,x3
Xi X2 X3 X n
~i b2-f2 b2-s +33 + ~ 2
X~l ~ 2 X3 + 1- 1 ' ' ' -..
x - _ (31-b )(3 1- 2 ).... (3 - b )
1 - 2 (/31- /2)...( 1-/ 3)'
= -_ (3 2-bb82-b2 ).... ((32-b)
(2- (-).... (/,- On)
the binomial factors of the numerator in the case of x, being
got by subtracting from /3r all the b's in succession, and the
similar factors of the denominator by subtracting from /3r all
the other [3's.
Remembering that Binet had originally been an expert in
working with determinants, it is not a little curious to note that
he did not compare with these expressions for x1, xI, x3,....
the expressions in terms of determinants, viz.x 1 = 1 (bl - )-)1.. (b -/,) + (b -12)-1 (b-.2)-.. (61 - )-n
1 (b2-32)-1... (b2-a)- (b-1)-1 (b- b1 ( )-'... (.b2-13)-1
1 (bn-13) )-.. (b-n)-1 (b-.) (b-,2)-1 * (bn n)
Had he done so he would undoubtedly have reached a result
which was not brought to light until four years later by Cauchy.