In the second edition the main plan of the book has been left unchanged. Small amounts of material have been added in a great number of places, and improvements have been attempted at many points. It was felt that a lack in the first edition was its omission of the theory of Laplace and Fourier transforms. This has been remedied in Chapter 8. The discussion of numerical calculations, integral equations and group theory has likewise .been augmented by removal of unnecessary items and some replacements.