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

148 HISTORY OF THE THEORY OF DETERMINANTS
The common form is that which we have seen obtained above by
Trudi as an equivalent for the quotient of an alternant by the
difference-product of the variables, and we thus conclude that the
given determinant must be equal to
K1 K2 Kn I -Pll,-11 - 0 zP1- n -
al1 2 an a +2 +(X-l
0 1 %?- 1 0 n-1
aI1 a2 a. a i I ay-^01 a 1.. n+-1
a... n a..
MERTENS, F. (1872).
[Auszug aus einem Schreiben.... Crelle's Journ., lxxv. p. 264.]
The matter here dealt with is Borchardt's generating function
of 1855 (Hist., ii. pp. 173-175). In its place Mertens suggests
n-1 n-2 0 I tot1 tn-1
t1 t2 2 -1 t... ti ' V (t l, t.... tn).f(tl) /(Q2)... (t)
wheref(t) = (t-al)(t-a2)... (t-a,), asserting that the coefficient
of (t1t... t,)-l in this is the symmetric function V(a,, a2,..., a).
As Borchardt's function had undeservedly withdrawn attention
from Jacobi's result of 1841 (Hist., i. pp. 336-339), namely, that the
symmetric function
90 (a, a 2, * a.)
is the coefficient of (tt2... t,)-l in
t0 I * I n (1, t1)
(- 1)2' l- f(t.). f(t)... f(t,
Mertens' closely analogous result has a double importance. In a
sense Mertens' theorem is more general than Jacobi's: for the
integral symmetric function, V, in the former is any such function,
whereas in the latter it is one which arises from the division of an
alternating function by the corresponding difference-product.
Taking advantage of this and of the fact that
(a,, a2, a-) (a, a2,...A, a)
nO1- 0 1 c4-1
tal(2~ a ~ 2 a '.. n
2 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~1