In a paper “On the Numerical Calculation of a Class of Definite Integrals and Infinite Series,” printed in the ixth volume of the Transactions of this Society, I gave a method by which a definite integral, to which Mr Airy was led in calculating the intensity of light in the neighbourhood of a caustic, may be readily calculated for large values, whether positive or negative, of a certain variable which appears as a constant under the sign of integration. The method consists in forming a differential equation of which the definite integral is a particular solution, obtaining the complete integral of the equation under a form, indicated by the equation itself, involving series according to descending powers of the variable, and determining the arbitrary constants. The equation admits also of integration by means of ascending series multiplied by other arbitrary constants. The ascending series are always convergent, but when the variable is large begin by diverging rapidly: the descending series, on the other hand, are always divergent, but when the variable is large begin by converging rapidly.

The same method was found to apply to several other definite integrals which occur in physical investigations, as well as to differential equations of frequent occurrence. The ascending and descending series are usually both required, the one for application to small, the other to large values of the variable; and it is necessary to connect the arbitrary constants in the descending with those in the ascending series.