THE present volume is devoted mainly to an investigation of the properties of the remarkable expressions which were first introduced to the notice of mathematicians by Legendre, and are now known as Laplace's Coefficients and Functions. Some account of these expressions is given in various works, but their importance in modern researches suggests the advantage of a more complete and systematic development of them than has hitherto appeared in England. The work now published will it is hoped be found sufficiently elementary for those who are commencing the subject, and at the same time adequate in extent to the wants of the advanced student.
The book is composed of four parts. The first part consists of twelve Chapters, in which the expressions are considered as functions of only a single variable; in this form they were first introduced by Legendre, and it is convenient to denote them, thus restricted, by his name. The second part consists of eight Chapters, in which the expressions are considered as functions of two variables; this is the form in which they present themselves in the writings of Laplace. The third part consists of nine Chapters which treat of Lamp's Functions; these maybe regarded as an extension of Laplace's Functions. The fourth part consists of seven Chapters which treat of Bessel's Functions; these are not connected with the main subject of the book, but as they are becoming very prominent in the applications of mathematics to physics it may be convenient to find an exposition of them here.
The demonstrations which are adopted have been carefully chosen so as to bring under the attention of students some of the most instructive processes of modern analysis. Thus the work may be regarded both as an account of the Functions to which it is specially devoted, and also as a continuation of the two volumes already published on the Differential and Integral Calculus respectively; the three together form a connected treatise on the higher department of pure mathematics....