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

ALTERNANTS (JOACIIIMSTHAL, 1854)

171

wo die rechte Seite eine Summe von 1)... n-a+1)
ahnlich gebildeten Glieder enthiilt." 1.2... a
The known result (Cayley's) obtained from this by equating
coefficients of x" is pointed out: also the extension
So Si 53..... 8(1,I
S1 S X S, x
S2 S S ". XS x E CJl2 6a 8. x( ), 2m..., Xa)
~2 S3 S4 *... Sa+1 X2 (X - X-1)(X - X2)....(X Xa)
Sa Sa+1 Sa+2.... S2a-1 Xa
where Si = x+~i +, ex' It may be noted that the
reason for discussing such determinants is that the series of them
obtained by giving a the values n, n-i, n-2,..., 2, 1, 0 is
put forward (p. 400) as a substitute for Sturm's series of
functions. -
Towards the end of the paper (~ 17, p. 414) the determinant
(a1+b1)-1 (a,+ b2j_.... (a., + bit)
(at+2+b1)-1 (a2+b2)-1. 0... (a+b,)- o
or A,, -say,
(a,,+b,) I- (an+ b2)-l.... (an,+ b,)-)
is evaluated. The process consists at the outset in subtracting
the first column from each column after the first, removing the
factor
(b1-b2)(b, - b3)... (b1, -bi)
(a, + bl9(a2 + bl)...(an + bd)
and writing the cofactor in the form
1 (a, +b2)'.... (a, +bi)1
1 (a2+ b2)-1...(a2+.
1 (an b2Ym.... (a,+bn)-l
The latter determinant is then transformed by subtracting the
* In this connection papers by Cayley (1846) and Borchardt (1845) are referred
to by Joachimsthal, but no mention is made of Sylvester's (1839).