portion of the subject. The present volume, therefore, even where its obligations to the great German philosopher are the deepest, is not a mere epitome of his work, but the result of independent study.

TRINITY COLLEGE, CAMBRIDGE,

June, 1873.

1 Die Lehre von den Tonempfindungen.

Dritte Ausgabe.

Braunschweig. 1870. Of this profound and exhaustive treatise it is not too much to say that it does for Acoustics what the Principia of Newton did for Astronomy.

PREFACE TO THE SECOND EDITION.

THE present edition is the result of a careful revision applied to its predecessor.

An unsatisfactory artificial hypothesis as to the motion of particles in the surface of water-waves, which was avowedly introduced into the first edition. for the sake of simplicity, is replaced by a mode of treatment based upon observed facts.

Points on which special difficulty was experienced by readers of the first edition are here somewhat more fully elucidated.

The author has, of course, consulted the revised edition' of Helmholtz's Tonempfindungen.

1

TRINITY COLLEGE, CAMBRIDGE,

June, 1883.

1 Vierte umgearbeitete Ausgabe. Braunschweig, 1877.

PREFACE TO THE THIRD EDITION.

THE present edition is substantially identical with its predecessor.

A reference to the chief acoustical discoveries of the late Professor von Helmholtz, which appeared on the title-page of the previous editions, has been omitted here. It is no longer called for, now that the eminently fruitful contributions to acoustical knowledge made by that great discoverer are definitively incorporated in the mass of our scientific heritage.

TRINITY COLLEGE, CAMBRIDGE,

December, 1895.

CONTENTS.

CHAPTER I.

ON SOUND IN GENERAL AND THE MODE OF ITS TRANSMISSION.

Sensation of Sound, and its cause, § 1-Connexion of Sound with

motion, § 2-Velocity of Sound, § 3-Stationary media of

Sound, § 4-Motion of sea-waves, § 5-Description of a wave,

§ 6-Length, amplitude and form of wave, § 7-Water-waves,

§ 8-Waves due to transverse vibrations, § 9-Periodic vibra-

tions, § 10-Form of wave and mode of vibration, §§ 11, 12, 13—

Wave on the surface of a field of standing corn, § 14-Longi-

tudinal vibrations, SS 15, 16-Condensation and rarefaction,

§ 17-Associated wave, § 18-Law of pendulum-vibration,

§ 19-Pressure and density of air; Mariotte's law, § 20-Trans-

mission of sonorous waves along a tube of uniform bore, § 21—

Unconstrained motion of Sound-waves, § 22-Musical and non-
musical sounds, § 23.

pp. 1-52

CHAPTER II.

ON LOUDNESS AND PITCH.

Three elements of a musical sound, loudness, pitch and quality,

§ 24-Loudness and extent of vibration, § 25-Pitch and rapidity

of vibration; the Syren; continuity of pitch, §§ 26, 27-Measure

of pitch; vibration-numbers, § 28-Limits of musical sounds,

§ 29-Relative pitch; intervals, § 30-Tonic intervals of the

Major scale; concords and discords, § 31-Additional notes

required for the Minor scale, § 32-Measure of intervals, § 33—

Vibration-fractions, § 34-Table of vibration-fractions for the

tonic intervals of the Major and Minor scales, § 35-Calculation

of the vibration-numbers of all the notes in a scale from the

vibration-number of its tonic, § 36.
pp. 53-73

CHAPTER III.

ON RESONANCE.

Resonance of pianoforte wires and tuning-forks; cause of the

phenomenon, $$ 37, 38-Resonance of a column of air; laws of

its production, § 39-Relation between the length of an air-

column and the pitch of its note of maximum resonance, § 40—

Resonance-boxes, § 41-Helmholtz's resonators, § 42.

CHAPTER IV.

ON QUALITY.

pp. 74-86

Composite nature of musical sounds in general; series of con-

stituent tones, and law which connects them, § 43—Experimental

analysis of musical sounds, §§ 44, 45-Nomenclature of the sub-

ject, § 46-Helmholtz's theory of musical quality as depending

on the number, orders and relative intensities of the partial-tones

present in any given clang, § 47.

CHAPTER V.

pp. 87-99

ON THE ESSENTIAL MECHANISM OF THE PRINCIPAL MUSICAL INSTRU-

MENTS, CONSIDERED IN REFERENCE TO QUALITY.

Sounds of tuning-forks, § 48-Modes of vibration of an elastic

tube, § 49-Meeting of equal and opposite pulses; formation of

nodes, § 50-Number of nodes formed, § 51-Nature and rate of

segmental vibration, §§ 52, 53-Motion of a sounding string;

quality and pitch of its note, §§ 54, 55-The pianoforte, § 56—

Meeting of equal and opposite systems of longitudinal waves,

§ 57-Reflexion of Sound at a closed and at an open orifice,

§ 58-Modes of segmental vibration in stopped and open pipes,

S$ 59, 60-Deepest note obtainable from a pipe, § 61-Relation

between length of pipe and pitch of note, § 62-Theory of re-

sonance-boxes, § 63-Flue-pipes and reed-pipes, § 64-Construc-

tion of flue-pipes and quality of their sounds, § 65-Mechanism

of a reed; quality of an independent reed, and of a reed associated

with a pipe, § 66-Orchestral wind-instruments, § 67-Mechan-

ism of the human voice, § 68-Synthetic confirmation of Helm-

holtz's theory of quality, § 69.
pp. 100-141