SUMMARY OF CHAPTER VII 374 CHAPTER VIII Law of Vibratory Motions in Water and Air—Superposition of Vibrations—Interference of Sonorous Waves—Destruction of Sound by Sound—Combined Action of Two Sounds nearly in Unison with each other—Theory of Beats—Optical Illustration of the Principle of Interference—Augmentation of Intensity by Partial Extinction of Vibrations—Resultant Tones—Conditions of their Production—Experimental Illustrations—Difference-Tones and Summation-Tones—Theories of Young and Helmholtz 377 SUMMARY OF CHAPTER VIII 407 CHAPTER IX Combination of Musical Sounds—The smaller the Two Numbers which express the Ratio of their Rates of Vibration, the more perfect is the Harmony of Two Sounds—Notions of the Pythagoreans regarding Musical Consonance—Euler’s Theory of Consonance—Theory of Helmholtz—Dissonance due to Beats—Interference of Primary Tones and of Overtones—Mechanism of Hearing—Schultze’s Bristles—The Otoliths—Corti’s Fibres—Graphic Representation of Consonance and Dissonance—Musical Chords—The Diatonic Scale—Optical Illustration of Musical Intervals—Lissajous’s Figures—Sympathetic Vibrations—Various Modes of illustrating the Composition of Vibrations 410 Summary of Chapter IX 450 APPENDIX I ON THE INFLUENCE OF MUSICAL SOUNDS ON THE FLAME OF A JET OF COAL-GAS. BY JOHN LE CONTE, M.D. 454 APPENDIX II ON ACOUSTIC REVERSIBILITY 461 INDEX 471 ILLUSTRATION—FOG-SIREN _Frontispiece_ [Illustration: FOG-SIREN] PREFACE TO THE THIRD EDITION In preparing this new edition of “Sound,” I have carefully gone over the last one; amended, as far as possible, its defects of style and matter, and paid at the same time respectful attention to the criticisms and suggestions which the former editions called forth. The cases are few in which I have been content to reproduce what I have _read_ of the works of acousticians. I have sought to make myself experimentally familiar with the ground occupied; trying, in all cases, to present the illustrations in the form and connection most suitable for educational purposes.