b2 b3 b_{n}
I. The infinite continued fraction -- -- ----- converges
a2 + a3 + ... + a_{n} + ...
to an incommensurable limit if after some finite value of n the condition
a_{n} [not <] b_{n} is always satisfied.
b2 b3 b_{n}
II. The infinite continued fraction -- -- -----
a2 - a3 - ... - a_{n} - ...
converges to an incommensurable limit if after some finite value of n
the condition a_{n} [>=] b_{n} + 1 is always satisfied, where the sign >
need not always occur but must occur _infinitely often_.
_Continuants._
The functions p_{n} and q_{n}, regarded as functions of a1, ..., a_{n},
b2, ..., b_{n} determined by the relations
p_{n} = a_{n}p_{n-1} + b_{n}p_{n-2},
q_{n} = a_{n}q_{n-1} + b_{n}q_{n-2},
with the conditions p1 = a1, p0 = 1; q2 = a2, q1 = 1, q0 = 0, have been
studied under the name of _continuants_. The notation adopted is
/ b2,...,b_{n}\
p_{n} = K ( ),
\a1, a2,...,a_{n}/
and it is evident that we have
/ b3,...,b_{n}\
q_{n} = K ( ).
\a2, a3,...,a_{n}/
The theory of continuants is due in the first place to Euler. The reader
will find the theory completely treated in Chrystal's _Algebra_, where
will be found the exhibition of a prime number of the form 4p + 1 as the
actual sum of two squares by means of continuants, a result given by H.
J. S. Smith.