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

CONTINUANTS (SYLVESTER, 1853)

4~23

ment" that anything apparently new in snbstance is met with.
There, in ~ a (p. 497), the following lemma occurs:"The roots of the eumulant [qj, q2,..., M], in which each element is
a linear function of x, and wherein the coefficient of x for each element
has the like sign, are all real: and between every two of such roots is
contained a root of the cumulant [q1, q2,..., qi,] and ex converso a root
of the cumulant [q2, q8,..., qj]: and (as an evident corollary) for all
values of ~ and (' intermediate between 1 and i the greatest root of
[q1, q,..., qi] will be greater, and the least root of the same will be
less than the greatest and least roots respectively of
[qP1 qp+1)... ) q' - 1 4 qP~
Even this, however, may be placed under the well-known theorem
regarding the roots of the equation
a11- (12 a13
CL12 22- a23 = 0
a153 a3 a33 - X
which had been enunciated by Cauchy in 1829.
The next noteworthy resnlt occupies S i. (p. 502). As a preparation for it the theorem
[a1, a2,..., am, bp, b2,.., baJ = [ca,, a2, a,,,][b,, b2,..., bz
- [a1, a2,..., aq-1] [b2, b3,.., bn]
may be recalled, the group of elements on the left being now
viewed as consisting of two sub-groups. This theorem Sylvester
wTrites in the form
WIN2] = [11][Q2] - [Q'1]Li221,
and he sncceeds in including it in a general theorem, not explicitly formulated, in which the number of groups is i, the next
two cases being
[S21Q2Q3] = [ [ [s231 [Q2 3]
- [ ['i22] [Q3] - [p1 [Q1 R['Q3] + [ ['Q1'] R1['Q3],
and
[Q1Q2Q3Q4] = [21 K[Q22] [R~3] [QJ
- 2 ] 'Q2] ["QJ ][2141 - [K21 [Q2] ['QJ 3[Q4 - [Q1][Q2] [QM [R'Q4]
+ WJF] ['2 i[Q3[124] + [+ K'11[',221K[]iM'24] + K21][M2 ]['F']1['I24]
- [QII ['Q2 1['Q21 [ ['23 j.