The underlying theme of this monograph is that the fundamental simplicity of the properties of orthogonal functions and the developments in series associated with them not only commends them to the attention of the student of pure mathematics, but also renders them inevitably important in the analysis of natural phenomena which lend themselves to mathematical description.

It is the essence of mathematics that it concerns itself with those relations which lie so deep in the nature of things that they recur in the most varied situations. This is particularly true, of course, of the rudimentary notions of arithmetic and geometry which have forced themselves on the attention of mankind since the earliest beginnings of thought. But with the advance of science and the accompanying extension of the range of phenomena subjected to quantitative discussion, more highly organized groups of concepts, gradually simplified by reduction to their essentials, have come to manifest themselves with similar persistence.

Among these are the formulations relating to the general analytical concept of orthogonality, and, in a more restricted field, the particular types of orthogonal systems discussed in the following chapters.

The choice of material is guided by two main lines of development, which are intimately associated from the beginning, but ultimately diverge in accordance with alternative principles of generalization.