The work here submitted to the notice of the public forms the third volume of a course intended to furnish to the mathematical student a pretty comprehensive view of the principles of modern analytical science. To complete this design will require a fourth volume, in some measure supplementary to the three now completed, and to contain the subject of Finite Differences, a fuller inquiry into the theory of Partial Differential Equations, and a chapter on Definite Integrals. This final volume I hope hereafter to be able to prepare, although I do not propose to enter immediately upon the undertaking.