∫l0 ρur(x) us(x) dx = 0.
(12)
This enables us to determine the coefficients, thus
Cr = ∫l0 ρƒ(x) ur (x)dx ÷ ∫10 ρ {ur(x)}² dx.
(13)
The extension to spaces of two or three dimensions, or to cases where there is more than one dependent variable, must be passed over. The mathematical theories of acoustics, heat-conduction, elasticity, induction of electric currents, and so on, furnish an indefinite supply of examples, and have suggested in some cases methods which have a very wide application. Thus the transverse vibrations of a circular membrane lead to the theory of Bessel’s Functions; the oscillations of a spherical sheet of air suggest the theory of expansions in spherical harmonics, and so forth. The physical, or intuitional, theory of such methods has naturally always been in advance of the mathematical. From the latter point of view only a few isolated questions of the kind had, until quite recently, been treated in a rigorous and satisfactory manner. A more general and comprehensive method, which seems to derive some of its inspiration from physical considerations, has, however, at length been inaugurated, and has been vigorously cultivated in recent years by D. Hilbert, H. Poincaré, I. Fredholm, E. Picard and others.
References.—Schuster’s method for detecting hidden periodicities is explained in Terrestrial Magnetism (Chicago, 1898), 3, p. 13; Camb. Trans. (1900), 18, p. 107; Proc. Roy. Soc. (1906), 77, p. 136. The general question of expanding an arbitrary function in a series of functions of special types is treated most fully from the physical point of view in Lord Rayleigh’s Theory of Sound (2nd ed., London, 1894-1896). An excellent detailed historical account of the matter from the mathematical side is given by H. Burkhardt, Entwicklungen nach oscillierenden Funktionen (Leipzig, 1901). A sketch of the more recent mathematical developments is given by H. Bateman, Proc. Lond. Math. Soc. (2), 4, p. 90, with copious references.
(H. Lb.)