The writing of this book was undertaken with the object of providing a text-book on the elements of the theory of the Spherical Harmonics, with applications to mathematical physics, so far as this could be done without employing the method of contour integration. Subsequently it was thought advantageous to include discussions on similar lines of Fourier Series and Bessel Functions, with corresponding applications.
The first chapter contains an elementary account of the theory of Fourier Series, while the second and third deal with the applications of Fourier Series to Conduction of Heat and Vibrations of Strings. The four following chapters form the central part of the book. In Chapter IV. the Spherical Harmonics are defined, and a summary is given of the elementary properties of the Hypergeometric Function. Chapters V, VI, and VII are devoted respectively to the Legendre Coefficients, the Legendre Functions, and the Associated Legendre Functions.
In Chapters VIII, IX, and X the Spherical Harmonics are employed to obtain expressions for the gravitational and electrostatic potentials of bodies bounded by circles, spheres, and spheroids; Chapters Xl and XII include similar discussions for bodies bounded by ellipsoids of revolution and eccentric spheres. A short account of Clerk Maxwell's theory of the Spherical Harmonics will be found in Chapter XIII. The remaining three chapters deal with the Bessel Functions and their applications to Vibrations of Membranes and Conduction of Heat.
At all stages of the work, as in the course of many previous undertakings, I have been indebted to Professor G A Gibson, LL.D., for important criticisms and valuable suggestions. To him my warmest thanks are due. I have also to thank my colleague, Mr William Arthur, M.A., for the great care with which he has read through all the proof sheets.
Among the books that proved useful to me, special mention should be made of the following: Wangerin's Theorie des Potentials und der Kugelfunktionen; Carslaw's Conduction of Heat; Schafheitlin's Theorie der Besselschen Funktionen; and Lamb's Dynamical Theory of Sound. I have also made use of lectures by Professor E W Hobson, F.R.S.
THOMAS M MACROBERT
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