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### Contents

 CHAPTER 1 CHAPTER II 19 Composition of harmonic motions of like period Harmonic Curve Com 36 CHAPTER IV 91 CHAPTER V 130 Cases in which the three functions T F V are simultaneously reducible 167 CHAPTER VI 170
 CHAPTER VII 242 CHAPTER VIII 255 i 292 CHAPTER IX 306 CHAPTER 395 PAGE 433

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Page 273 - K — the radius of gyration of the section about an axis perpendicular to the plane of bending— and inversely as the square of the length. These results might have been anticipated by the argument from dimensions, if it were considered that the frequency is necessarily determined by the value of...
Page 389 - MOTION. 389 excited at points distant therefrom 0, 90, 180, or 270 degrees.; and normally when the friction was applied at the intermediate points corresponding to 45, 135, 225 and 315 degrees. Care is sometimes required in order to make the bell vibrate in its gravest mode without sensible admixture of overtones. If there be a small load at any point of the circumference, a slight augmentation of period ensues, which is different according as the loaded point coincides with a node of the normal...
Page 109 - The period of a conservative system vibrating in a constrained type about a position of stable equilibrium is stationary in value when the type is normal. We might prove this from the original equations of vibration, but it will be more convenient to employ the normal co-ordinates. The constraint, which may be supposed to be of such a character as to leave only one degree of freedom, is represented by taking the quantities <f> in given ratios.
Page 23 - Where two notes differing slightly in pitch form beats, the number of beats in a second is equal to the difference of the frequencies. If, then, the frequencies of the notes sounded differ by one vibration per second nearly, there will be one beat per second ; if they differ by two or three per second there will be two or three beats. The frequencies of the two lowest notes of the 32-foot range are sufficiently nearly...
Page 182 - ... (1) For a given string and a given tension, the time varies as the length. This is the fundamental principle of the monochord, and appears to have been understood by the ancients1. (2) When the length of the string is given, the time varies inversely as the square root of the tension. (3) Strings of the same length and tension vibrate in times, which are proportional to the square roots of the linear density. These important results may all be obtained by the method of dimensions, if it be assumed...
Page 129 - We may readily prove from this that in order to deduce the motion depending on initial displacements from that depending on the initial velocities, it is only necessary to differentiate with respect to the time, and to replace the arbitrary constants (or functions) which express the initial velocities by those which express the corresponding initial displacements. Thus, if...
Page 170 - A' as much as possible, the corresponding value of a' is one half of its maximum. CHAPTER VI. TRANSVERSE VIBRATIONS OF STRINGS. 118. AMONG vibrating bodies there are none that occupy a more prominent position than Stretched Strings. From the earliest times they have been employed for musical purposes, and in the present day they still form the essential parts of such important instruments as the pianoforte and the violin. To the mathematician they must always possess a peculiar interest as the battle-field...
Page 475 - ... to represent the crests of the component trains. In the case of two trains of slightly different wave-lengths, it may be proved that the tangent of the angle between the line of maxima and the wave-fronts is half the tangent of the angle between the wave-fronts and the boat's course.
Page 109 - The reduction to-normal co-ordinates allows us readily to trace what occurs when two of the values of n1 become equal. It is evident that there is no change of form. The spherical pendulum may be referred to as a simple example of equal roots. It is remarkable that both Lagrange and Laplace fell into the error of supposing that equality among roots necessarily implies terms containing t as a factor1. The analytical theory of the general case (where the co-ordinates are not normal) has been discussed...
Page 475 - ... into still water the velocity of the group is less than that of the individual waves of which it is composed ; the waves appear to advance through the group, dying away as they approach its anterior limit. This phenomenon seems to have been first explained by Prof.