The advent of high speed digital computers and the consequent intensification of interest in the study of numerical analysis has caused considerable attention to be paid to the problem of obtaining approximation formulas for functions which occur in the theory of mathematical physics. It is the purpose of this note to describe the theory underlying various methods of obtaining rational approximations to functions which are formally defined by a power series expansion; it is assumed that the power series concerned are quite general in character, and that the functions with which they are associated do not satisfy a particular functional equation which would permit the use of any special method. The theory is then subjected to a detailed analysis in terms of the computational steps involved, and a comparison, with regard to computational efficiency, of the various methods which may be devised for obtaining rational approximations is given.