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

CONTINUANTS (MUIR, 1874)

411

MUIR, T. (1874).
[Continuants: a new special class of determinants. Proceed. R. Soc.
Edinburgh, viii. pp. 229-236.]
Still another discoverer, twenty years after the publication of
Sylvester's first paper on the subject! Fortunately the' discoveries'
are not expounded at inordinate length, the paper consisting of
twenty-two very short paragraphs, almost every one of which
contains a concisely stated property, accompanied, where necessary,
by an indication of the mode of proof.
'Continuant' is formally defined as being a determinant which
has the elements lying outside the principal diagonal and the two
bordering minor diagonals each equal to zero, and which has the
element of one of these minor diagonals each equal to negative
unity. The continuant
al b1l
-1 a2 b2.. is denoted by K( bl b2
-1 a3.... a a2
a, a, a,... being spoken of as the main diagonal, and
b, b,... as the minor diagonal. When the b's are each equal
to unity, the continuant is called a simple continuant, and denoted
by K (a, a2, Is,... )
Of results previously formulated it may be worth noting that the
most important, namely,
K( al b... bah.K(... bpl where h<p<n
K K(..... bp. K ( b.... b.
is not obtained in the same manner as in Thiele's paper of 1870, a
is not obtained in the same manner as in Thiele's paper of 1870,
but from a general theorem regarding the product of any determinant by one of its own minors.*
* This theorem was not published until five years afterwards. See above, p. 79.