Ernest William Hobson (1856-1933) was a prominent English mathematician who held the position of Sadleirian Professor at the University of Cambridge from 1910 to 1931. In this volume, which was originally published in 1931, Hobson focuses on the forms and analytical properties of the functions which arise in connection with those solutions of Laplace's equation which are adapted to the case of particular boundary problems. The investigations take into account functions not, as was the case when they were originally introduced, confined to the cases where degree and order are integral. This is a highly informative book that will be of value to anyone with an interest in spherical and ellipsoidal harmonics.