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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1893 Excerpt: ... am is a root of the equation ac cos ac + p sin ac = 0. (13) It is to be observed that if p is infinite (13) reduces to sinac = 0, am becomes and (11) and (12) give our regulation Fourier sine series (v. Art. 31), and therefore the ordinary Fourier development in sine series is merely a special case of the problem just solved. Moreover since the Fourier method of determining the coefficients of such a series requires that e I sin amx sin anx.dx = 0, 0 that is that -s naa +aan)C = or reducing, that 54 =, sin amc sin anc or that am and an should be roots of the equation ac cos ac _ sin ac where p is some constant, it follows that we have obtained in (11) the most general sine development that can be obtained by Fourier's method. EXAMPLES. 1. Show that the solution of the problem of Art. 67 is 2. If the initial temperature of the sphere is constant and equal to am ' ac2 + Ac (Ac--1) 3. If the temperature of the air is a constant instead of zero the surface equation of condition is Dru--h (u--y)=0 when r = c. The substitution of t = u--, however, brings the problem under Ex. 1 and we get where am has the value given in Ex. 6. 69. If we have an infinite solid with one plane face which is exposed to air at the temperatures U=F(t) and heat can flow only at right angles to this face, we can solve the problem readily for the case where the initial temperatures are zero. We have Determining by the fact that uehx = 0 when x = oo we have 00 u = hehx Ce hx vdx. (3) X Substituting the value of v from (2) we have as our required solution. For an extension of this method to the flow of heat in two and three dimensions and for the interpretation of the results by the aid of the theory of...

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