## An Elementary Treatise on Fourier's Series and Spherical, Cylindrical, and Ellipsoidal Harmonics: With Applications to Problems in Mathematical Physics |

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axis coefficients convergent convergent series cosh cosines Crelle's Journal curve curvilinear coordinates cylinder degree differential equation doublet of strength E. W. Hobson ellipsoid Ellipsoidal Harmonics equal to zero EXAMPLES expressed external point find the temperature finite flow of heat formula Fourier's Series give given function Green's Theorem Hence homogeneous homogeneous function indefinitely increased initial temperature initially distorted Lame's Laplace's Equation Laplacian Legendre's Equation limiting value lines of flow membrane method mirx multiply mx.dx obtained odd function parallelopiped particular solutions plane Pm(x positive integer potential function due radius rectangular reduces required solution roots satisfy second member Show sin x sin2 sinh slab solid solution of Laplace's solution of Legendre's spherical coordinates spheroid string substitute Surface Spherical Harmonic Surface Zonal Harmonic symmetrical temperature zero Trigonometric Series unit sphere vibration whence whole number

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Page 230 - ... is set up. Find the temperature at points on the axis 25 cm., 50 cm., and 75 cm. from the base, and also at a point 25 cm. from the base and 50 cm. from the axis.

Page 226 - APPLICATIONS OF BESSEL'S FUNCTIONS. (0) The problem of Art. 5 ¡sa special case of the following : The convex surface and one base of a cylinder of radius a and length b are kept at the constant temperature zero, the temperature at each point of the other base is a given function of the distance of the point from the center of the base ; required the temperature of any point of the cylinder after the permanent temperatures have been established. Here we have to solve Laplace's Equation in the form...

Page 106 - ... 100° throughout, are placed together face to face, and their outer faces are kept at the temperature 0°. Find the temperature of a point in their common face and of points 10 cm. from the common face fifteen minutes after the slabs have been put together. Given

Page 47 - L.etf(x) be the given function of x. It can be expressed as the sum of an even function of x and an odd function of x by the following device...

Page 12 - ... make the discussion of equation (1) as straightforward as possible, let us first carry through the treatment in a formal manner and not be concerned with the specific details of the functions and symbols which may appear. II...

Page 19 - Let these roots be called a and /S, then is a solution, and since it contains two arbitrary constants it is the general solution.

Page 274 - ... The Physician as a Naturalist " (1888) ; and many papers in medical journals, and in the transactions of pathological and medical societies. ERNEST WILLIAM HOBSON, D.Sc. (Cantab.). Fellow of Christ's College, Cambridge, and University Lecturer. Author of the following memoirs, paper and book: — "On a Class of Spherical Harmonics of Complex Degree with Applications to Physical Problems

Page 269 - An essay on the application of mathematical analysis' to the theories of electricity and magnetism, Nottingham 1828.

Page 62 - The preceding figures represent the first four approximation to this curve. In each figure the curve y = the series, and the approximations in question are drawn in continuous lines, and the preceding approximation and the curve corresponding to the term to be added are drawn in dotted lines. Prob. n. Construct successive approximations to the series given in the examples at the end of Art. 6. Prob. 12. Construct successive approximations to the Maclaurin's x3 x...

Page 268 - Fourier was the first to assert and to attempt to prove that any function, even though for different values of the argument it is expressed by different analytical formulae, can be developed in such a series.