This enables us to determine the coefficients, thus
| (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.)
HARMONICHORD, an ingenious kind of upright piano, in
which the strings were set in vibration not by the blow of the
hammer but by indirectly transmitted friction. The harmonichord,
one of the many attempts to fuse piano and violin, was
invented by Johann Gottfried and Johann Friedrich Kaufmann
(father and son) in Saxony at the beginning of the 19th century,
when the craze for new and ingenious musical instruments was
at its height. The case was of the variety known as giraffe.
The space under the keyboard was enclosed, a knee-hold being
left in which were two pedals used to set in rotation a large
wooden cylinder fixed just behind the keyboard over the levers,
and covered with a roll-top similar to those of modern office
desks. The cylinder (in some specimens covered with chamois
leather) tapered towards the treble-end. When a key was
depressed, a little tongue of wood, one end of which stopped the
string, was pressed against the revolving cylinder, and the
vibrations produced by friction were transmitted to the string
and reinforced as in piano and violin by the soundboard. The
adjustment of the parts and the velocity of the cylinder required
delicacy and great nicety, for if the little wooden tongues rested
too lightly upon the cylinder or the strings, harmonics were
produced, and the note jumped to the octave or twelfth. Sometimes
when chords were played the touch became so heavy that
two performers were required, as in the early medieval organistrum,
the prototype of the harmonichord. Carl Maria von
Weber must have had some opinion of the possibilities of the
harmonichord, which in tone resembled the glass harmonica,
since he composed for it a concerto with orchestral accompaniment.
(K. S.)
HARMONIUM (Fr. harmonium, orgue expressif; Ger. Physharmonika,
Harmonium), a wind keyboard instrument, a small
organ without pipes, furnished with free reeds. Both the
harmonium and its later development, the American organ, are
known as free-reed instruments, the musical tones being produced
by tongues of brass, technically termed “vibrators” (Fr.
anche libre; Ger. durchschlagende Zunge; Ital. ancia or lingua
libera). The vibrator is fixed over an oblong, rectangular frame,
through which it swings freely backwards and forwards like a
pendulum while vibrating, whereas the beating reeds (similar to
those of the clarinet family), used in church organs, cover the
entire orifice, beating against the sides at each vibration. A
reed or vibrator, set in periodic motion by impact of a current
of air, produces a corresponding succession of air puffs, the
rapidity of which determines the pitch of the musical note.
There is an essential difference between the harmonium and the
American organ in the direction of this current; in the former
the wind apparatus forces the current upwards, and in the latter
sucks it downwards, whence it becomes desirable to separate in
description these varieties of free-reed instruments.
 |
| By courtesy of Metzler & Co. |
| Fig. 1.—Free Reed Vibrator, Alexandre Harmonium. |
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
