where a1, a2, a3, ... and b2, b3, b4, ... are any quantities whatever, positive or negative, is called a "continued fraction." The quantities a1 ..., b2 ... may follow any law whatsoever. If the continued fraction terminates, it is said to be a terminating continued fraction; if the number of the quantities a1 ..., b2 ... is infinite it is said to be a _non-terminating_ or _infinite_ continued fraction. If b2/a2, b3/a3 ..., the _component fractions_, as they are called, recur, either from the commencement or from some fixed term, the continued fraction is said to be _recurring_ or _periodic_. It is obvious that every terminating continued fraction reduces to a commensurable number. The notation employed by English writers for the general continued fraction is b2 b3 b4 a1 +- -- -- -- ... a2 +- a3 +- a4 +- Continental writers frequently use the notation b2 b3 b4 b2 | b3 | b4 | a1 +- -- +- -- +- -- +- ..., or a1 +- |----| +- |----| +- |----| +- ... a2 a3 a4 | a2 | a3 | a4 The terminating continued fractions b2 b2 b3 b2 b3 b4 a1, a1 + --, a1 + -- --, a1 + -- -- --, ... a2 a2 + a3 a2 + a3 + a4 reduced to the forms a1 a1a2 + b2 a1a2a3 + b2a3 + b2a1 --, ---------, --------------------, 1 a2 a2a3 + b3 a1a2a3a4 + b2a3a4 + b3a1a4 + b4a1a2 + b2b4 ------------------------------------------, ... a2a3a4 + a4b3 + a2b4 are called the successive convergents to the general continued fraction. Their numerators are denoted by p1, p2, p3, p4...; their denominators by q1, q2, q3, q4.... We have 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}. b2 b3 b4 In the case of the fraction a1 - -- -- -- ..., we have the a2 - a3 - a4 - 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}. Taking the quantities a1 ..., b2 ... to be all positive, a continued b2 b3 fraction of the form a1 + -- -- ... is called a _continued fraction a2 + a3 + b2 b3 b4 of the first class_; a continued fraction of the form -- -- -- ... a2 - a3 - a4 - called a _continued fraction of the second class_. 1 1 1 A continued fraction of the form a1 + -- -- -- ..., where a2 + a3 + a4 +