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

326 HISTORY OF THE THEORY OF DETERMINANTS
that in Borchardt's paper of 1845 (January) a determinant
of the special form we are now considering appeared as an
expression for the square of the difference-product, and that
a generalisation of this result was given by Cayley the year
following. These two papers as well as four others dealt with
under Alternants should be kept in view in reading the present
chapter. The full list is1845 Borchardt, C. W., p. 159. 1854 Brioschi, F., p. 172.
1846 Cayley, A., p. 162. 1857 Bellavitis, G., p. 181.
1854 Joachimsthal, F., p. 169. 1847 Baltzer, R. p. 183.
JACOBI, C. G. J. (1845, August).
[Ueber die Darstellung einer Reihe gegebner Werthe durch eine
gebrochne rationale Function. Crelle's Journ., xxx. pp.
127-156; or Gesammelte Werkte, iii. pp. 479-511.]
The subject here dealt with by Jacobi is that first considered
by Cauchy in the fifth note to the Analyse Algebrique of 1821,
namely, the extension of Lagrange's interpolation-formula, or
the finding of a function u of the form N(x)/M(x) which shall
have the values ul, ZU2..., U,++1l when x has the values
x1, X2,,., ax+m, it being understood that N and M are
respectively of the nth and rnth degrees in x.
The given n + m + 1 equations
UM(X1) = N(x1), M(x)= N(......
are first used to eliminate the n+1 coefficients of N(x), and
thereby obtain nz equations for the determination of the ratios
of the coefficients of M(x). This is interestingly accomplished
by using the multipliers xP//f'(x1 /2), '()..... where
f(x) = (x-x)(x-x2).... (x-x~+m++), then performing addition,
and finally utilising a known theorem regarding "partial
fractions." The result is that for any one value of p we have
i=n7+7t+1 i=+m1+l
Z /x'M(x,) - xZ N(x) '
PX0>P(i