This enables us to determine the coefficients, thus

${\displaystyle C_{r}=\int _{0}^{l}\rho f(x)u_{r}(x)dx{\text{ ÷}}\int _{0}^{l}\rho \{u_{r}(x)\}^{2}dx.}$

(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.) 

The harmonium has a keyboard of five octaves compass when complete, and a simple action controlling the valves, &c. The necessary pressure of wind is generated by bellows worked by the feet of the performer upon foot-boards or treadles. The air is thus forced up the wind-trunks into an air-chamber called the wind-chest, the pressure of it being equalized by a reservoir, which receives the excess of wind through an aperture, and permits escape, when above a certain pressure, by a discharge valve or pallet. The aperture admitting air to the reservoir may be closed by a drawstop named “expression.” The air being thus cut off, the performer depends for his supply entirely upon the management of the bellows worked by the treadles, whereby he regulates the compression of the wind. The character of the instrument is then entirely changed from a mechanical response to the player’s touch to an expressive one, rendering what emotion may be communicated from the player by increase or diminution of sound through the greater or less pressure of wind to which the reeds may be submitted. The drawstops bearing the names of the different registers in imitation of the organ, admit, when drawn, the wind from the wind-chest to the corresponding reed compartments, shutting them off when closed. These compartments are of about two octaves and a half each, there being a division in the middle of the keyboard scale dividing the stops into bass and treble. A stop being drawn and a key pressed down, wind is admitted by a corresponding valve to a reed or vibrator (fig. 1). Above each reed in the so-called sound-board or pan is a channel, a small air-chamber or cavity, the shape and capacity of which have greatly to do with the colour of tone of the note it reinforces. The air in this resonator is highly compressed at an even or a varying pressure as the expression-stop may not be or may be drawn. The wind finally escapes by a small pallet-hole opened by pressing down the corresponding key. In Mustel and other good harmoniums, the reed compartments that form the scheme of the instrument are eight in number, four bass and four treble, of three different pitches of octave and double octave distance. The front bass and treble rows are the “diapason” of the pitch known as 8 ft., and the bourdon (double diapason), 16 ft. These may be regarded as the foundation stops, and are technically the front organ. The back organ has solo and combination stops, the principal of 4 ft. (octave higher than diapason), and bassoon (bass) and oboe (treble), 8 ft. These may be mechanically combined by a stop called full organ. The French maker, Mustel, added other registers for much-admired effects of tone, viz. “harpe éolienne,” two bass rows of 2 ft. pitch, the one tuned a beat too sharp, the other a beat too flat, to produce a waving tremulous tone that has a certain charm; “musette” and “voix celeste,” 16 ft.; and “baryton,” a treble stop 32 ft., or two octaves lower than the normal note of the key. The “back organ” is usually covered by a swell box, containing louvres or shutters similar to a Venetian blind, and divided into fortes corresponding with the bass and treble division of the registers. The fortes are governed by knee pedals which act by pneumatic pressure. Tuning the reeds is effected by scraping them at the point to sharpen them, or near the shoulder or heel to flatten them in pitch. Air pressure affects the pitch but slightly, being noticeable only in the larger reeds, and harmoniums long retain their tuning, a decided advantage over the organ and the pianoforte. Mechanical contrivances in the harmonium, of frequent or occasional employment, besides those already referred to, are the “percussion,” a small pianoforte action of hammer and escapement which, acting upon the reeds of the diapason rows at the moment air is admitted to them, gives prompter response to the depression of the key, or quicker speech; the “double expression,” a pneumatic balance of great delicacy in the wind reservoir, exactly maintaining by gradation equal pressure of the wind; and the “double touch,” by which the back organ registers speak sooner than those of the front that are called upon by deeper pressure of the key, thus allowing prominence or accentuation of certain parts by an expert performer. “Prolongement” permits selected notes to be sustained after the fingers have quitted