He was also the first to consider the difficult problems involved in equations of mixed differences, and to prove that an equation in finite differences of the first degree and the second order might always be converted into a continued fraction.
The exact determination of the appearances in any given case is a mere problem of convergents to a continued fraction.
If the continued fraction terminates, it is said to be a terminating continued fraction; if the number of the quantities a, ..., b 2 ..
If b 2 /a 2, 3 /a 3 ..., 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 al b2 b3 b4 a 2 "' Continental writers frequently use the notation a 1 ?
