There are a great many problems in Heat, Electricity, Fluid Motion, &c, the solution of which is effected by developing an arbitrary function, either in a series or in an integral, by means of functions of known form. The first example of the systematic employment of this method is to be found in Fourier's Theory of Heat. The theory of such developements has since become an important branch of pure mathematics.

Among the various series by which an arbitrary function f(x) can be expressed within certain limits, as 0 and a, of the variable x, may particularly be mentioned the series which proceeds according to sines of πx/a and its multiples, and that which proceeds according to cosines of the same angles. It has been rigorously demonstrated that an arbitrary, but finite function of x, f (x),may be expanded in either of these series. The function is not restricted to be continuous in the interval, that is to say, it may pass abruptly from one finite value to another; nor is either the function or its derivative restricted to vanish at the limits 0 and a. Although however the possibility of the expansion of an arbitrary function in a series of sines, for instance, when the function does not vanish at the limits 0 and a, cannot but have been contemplated, the utility of this form of expansion has hitherto, so far as I am aware, been considered to depend on the actual evanescence of the function at those limits.