REV. N. M. FERRERS, M.A., F.R.S., FELLOW AND TUTOR OF GONVILLE AND CAIUS COLLEGE, CAMBRIDGE. London: MACMILLAN AND CO. 1877 [All Rights reserved.] PREFACE. THE object of the following treatise is to exhibit, in a concise form, the elementary properties of the expressions known by the name of Laplace's Functions, or Spherical Harmonics, and of some other expressions of a similar nature. I do not, of course, profess to have produced a complete treatise on these functions, but merely to have given such an introductory sketch as may facilitate the study of the numerous works and memoirs in which they are employed. As Spherical Harmonics derive their chief interest and utility from their physical applications, I have endeavoured from the outset to keep these applications in view. I must express my acknowledgments to the Rev. C. H. Prior, Fellow of Pembroke College, for his kind revision of the proof-sheets as they passed through the press. GONVILLE AND CAIUS COLLEGE, N. M. FERRERS. F. H. Ъ CONTENTS. 1. Differential Equation of Zonal Harmonics. 2. General solution of this equation 3. Proof that P, is the coefficient of h' in a certain series 12. Expression of P, in ascending powers of μ 17. Expression of in a series of zonal harmonics |