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

180 HISTORY OF THE THEORY OF DETERMINANTS

or
( i- )n('1) (aC, g,.., a)2. ((a,1 n+l)(a2' +n+l) ~ ~ (al,+f '+1).
The result of the whole change was therefore
1 = ~'zx%, (-1),~
\ A(a,, a2..*., ay)
whence it followed that
= (-1)z (a C, 2,..., an);
and so the longed-for result was reached
J = (_)nA(a, a,,.. an) A(Ul,...., +)[1 2,,+1]
A1A2.. A,
Thereupon additional results came with ease. First we are
told that in a similar manner the determinant got from J by
changing the second power in the denominator of every element
into the first power* is found equal to
n (ai, a2 * * * n) a(iU U2... 2 U, l)
( A1A... A,
Then "E combinatione aequationum prodit
det. % 4 '1 1 '
(a1 + (a2+ )2' ' ( an + U[.
det. a+U a+U an+
Ut=-1, =t2, =.. =-n+1
Faciendo un+, = quantitati infinite magnae, aequatio in relationem a cl. Borchardt inventam transit, scilicet in
det. 1 1
et (a, +iU)21 (a2~m)2' ' "(an,+~U)2 1
det. 1 1 1
1 al+ a2+U " n+U
==ZG1 =U25 = * =Itn5
* Previous suggestions of such a determinant appear in Binet's paper of 1837
and Joachimsthal's of 1854.