The facilities which are afforded by the introduction of the function into certain classes of Differential expressions are well known. This function has effected the combination and generalization of Problems, which, although found to be capable of solution in particular cases, were regarded rather as isolated and exceptional forms, than as integral parts of some comprehensive expression. But the subject is far from exhausted. Some of the most important Differential Equations have never as yet been solved by a general method. The present Memoir is intended to supply this defect. The process employed differs little from that which I have previously exhibited; but the range of Problems which it embraces is much more extensive, and the Problems themselves are of a more important character.