There is an important class of functions, often occurring in physical investigations, whose numerical calculation is easy when the argument is either small or great. In the first case the function is readily calculated from an ascending series, which is always convergent and might be employed whatever the value of the variable may be, were it not for the length to which the calculations would run. When the argument is great, a series proceeding by descending powers is employed, whose character is quite different. In this case the series is of the kind called semi-convergent, though strictly speaking it is not convergent at all; for, when carried sufficiently far, the sum of the series may be made to exceed any assignable quantity. But, though ultimately divergent, it begins by converging, and when a certain point is reached the terms become very small. It can be proved that, if we stop here, the sum of the terms already obtained represents the required value of the functions, subject to an error which in general cannot exceed the last term included. Calculations founded on this series are therefore only approximate; and the degree of the approximation cannot be carried beyond a certain point. If more terms are included, the result is made worse instead of better. In the class of functions referred to, the descending series is abundantly adequate when the argument is large, but there will usually be a region—often the most interesting part of the whole— where neither series is very convenient. The object of the present note is to point out how a part of the difficulty thence arising may sometimes be met.