"2nd. We first perceive that this expression contains an infinite number of terms, into which unknown constants enter, or that it is equal to an which includes one or more arbitrary functions. In the first instance, [i.e.], when the general term is affected by the symbol \textstyle \sum , we derive from the special conditions a definite , whose roots give the values of an infinite number of constants. The second instance... when the general term becomes... infinitely small... the sum of the series is... changed into a definite integral."
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The Analytic Theory of Heat
The Analytic Theory of Heat (1878) is a translation by Alexander Freeman, M.A., with notes, of Joseph Fourier's Théorie Analytique de la Chaleur (1822). Fourier based his reasoning on : the flow of heat between two adjacent molecules is proportional to the extremely small difference of their temperatures. In this work Fourier claims that any function of a variable, can be expanded in a series of sines of multiples of the variable. Though not correct without additional conditions, Fourier's obser
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