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

340 HISTORY OF THE THEORY OF DETERMINANTS

and, as Trudi proves that this can only hold when the coefficients
of A and B vanish, it follows that each of the two series of
functions
Uo, U1,.. Unl, T0,,V,.., Vn,+
has one of the Sturmian properties which the p's have been shown
to possess.
As regards the highest-common-divisor (pp. 142-144) his result is:
In order that two functions may have a common divisor of the kth degree,
it is necessary and sufficient that the first determinant of each of the
last k of their bigradient arrays shall vanish: and, when this holds,
the coefficients of the divisor in question are the successive determinants
of the (n-k)t' array. For example, the functions being
aoxs + ax7+... a8, box5+bx... + 5,
their bigradient arrays are
(a,..., a8) (a,..., a8)2 (0,.., a )83
| (boa..., b5), (bo0,. *., ), (bo...,5)
(aO., a.8), (a,..., a8)5
(bo0,.. b5) (bo...,,b5),
and the proposition states that if the first determinant of each of
the last three arrays vanishes, the functions have the common
cubic factor
(cO, a )2 (x3, x2, x, 1).
(b,o *.., b5)
At a later stage (p. 151) there is given the supplementary proposition that the quotients resulting from dividing A and B by the said
highest-common-divisor are, save for an unimportant factor in each
case, the coefficients of B and A in Trudi's form of the (n - k + 1)th
'simplified remainder '-that is to say, are Vnk+1 and U7_,,l+ as
before defined.
The closely related question concerning the common roots of two
equations he deals with at length in a section devoted to elimination
(pp. 161-178). Starting with the proposition that, u and v being
integral functions of x, uA+vB must vanish for any common root
of the equations A = 0, B = 0, he next points out that u and v may
be so chosen as to make uA-vB of a low degree in x, even of the
degree zero. In the latter extreme case uA+vB must contain the