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

362 HISTORY OF THE THEORY OF DETERMINANTS

CAYLEY, A. (1848, August).
[Nouvelles recherches sur les fonctions de M. Sturm. Journ. (de
Liouvitle) de Math., xiii. pp. 269-274; or Collected Math.
Papers, i. pp. 392-396.]
Recalling his former paper on the same subject in Liouville's
Journal, xi. (1846), pp. 297-299, where Sturm's functions had
been expressed in terms of sums of powers of the roots of the
original function, he intimates now the discovery of more simple
expressions in terms of the coeficients of the said function. At
the same time he draws attention to the fact that his result may
be viewed as unconnected with Sturm's division-process, and it is
in this general light that he prefers to state it. Beginning with
two functions V and V' of the nth degree, namely,
(ax' + bx1"- +.... aIX + b''-l +....
and forming therefrom the series of functions
V V V V xV' V' ' x'V V V 2V' xV' V'
a a', a. a... Ca
b a b' a' b. b' a
c b c' b', c b a c' b ca'
cd c b c' c' b'
e c c e' cl' '
etc., which he denotes by -F1, F2, -F3,...., he affirms that
there is a homogeneous linear relation connecting every consecutive three of the latter functions,* namely,
PF3 + (xPlP2+PlP2+PlP2)F2 + PF1 = 0,
P2F4PP )F + P(+P, + P F, = 0,

* The signs require verification.