Colloquium publications.

138

THE BOSTON COLLOQUIUM.

mants are the successive convergents. * Thus a countless manifold
of continued fractions can be formed, any one of which through
its convergents gives the initial series to any required number of
terms and hence defines the series and table uniquely. In all of
Padd's continued fractions the partial numerators are monomials
in x.
The continued fraction is called regular when its partial numerators are all of the same degree and likewise its denominators,
certain specified irregularities being admitted in the first one or
two partial fractions. These irregularities disappear when the
continued fraction, as is most usual, commences with the corner
element of the table. (Cf. the continued fractions (2) and (3).)
In a normal table a regular continued fraction can be obtained
in any one of three ways. If we take for the convergents the
approximants which fill a horizontal or vertical line, a continued
fraction is obtained which-except for the irregularity permitted
at the outset-is of the form (1) given above. If the approximants lie upon the principal diagonal or any parallel line, the continued fraction is of type (3). Lastly, if the convergents lie upon
a stair-like line, proceeding alternately one term horizontally to
the right and one term vertically downward, the continued fraction
is of the familiar form (2).
When a table is not normal, the approximants which are identical with one another are shown by PadC to fill always a square,
the edges of which are parallel to the borders of the table. When
the square contains (n + 1)2 elements, the irregularity may be said
to be of the nth order. The vertical, horizontal, diagonal and
stair-like lines give regular continued fractions as before, unless
they cut into one or more of these square blocks of equal approximants. When this happens, certain irregularities appear in the
continued fraction which give rise to various difficulties in the
consideration of matters of convergence and other questions.
On this account it is natural to inquire first whether the continued fraction has or has not a normal character. If it has, the
* This is also tacitly implied in the relations given by Frobenius [13, p. 5].