In three papers communicated at different times to the Society, I have considered the application of divergent series to the actual and easy calculation, to an amply sufficient degree of accuracy, of certain functions which occur in physical investigations, but which can of course be considered quite apart from their applications. These functions present themselves as the complete integrals of certain linear differential equations, or it may be as definite integrals which lead to such differential equations, of which they form particular integrals; and as of course the theory of the complete integrals includes that of any particular integrals, the subject is best regarded from the former, and more general, point of view.

The independent variable was taken as in general a mixed imaginary, and the complete integral was expressed in two ways, either by ascending series which were always convergent, or by exponentials multiplied by descending series which were always divergent (except in very special cases in which they might terminate), though when the divergent series were practically useful they were of the kind that has been called semi-convergent. In either form of the complete integral, the arbitrary constants appeared as multipliers of the infinite series (of the ascending or descending as the case might be), or it might be, in part, of a function in finite terms.