A number of pulleys, one for each simple harmonic component, are carried by cranks which are made to revolve by clockwork, the length of each crank being proportional to the amplitude of the corresponding component. The axles are ranged in two rows, an upper and a lower, and a flexible wire passes alternately below a lower and above an upper pulley. The distances of the FIG. 40. upper from the lower pulleys are very great compared with the lengths of the cranks, and the portions of wire which run from each pulley to the next are always nearly vertical. One end of the wire is fixed, and the other carries a heavy pen, which is guided to move only in one vertical line. The pen keeps the wire tight, and is raised and lowered by the movements of the pulleys. If only one pulley moved, it would give the pen a simple harmonic motion, the amplitude of which would be double the length of the crank, because the height of the pulley determines the lengths of two of the free portions of wire. Hence, when all the pulleys are moving, the motion of the pen is the resultant of as many simple harmonic motions as there are pulleys. The axles are all driven by the same clockwork, a system of toothed wheels being employed to give them approximately the correct velocity-ratios. 96. The first machine of this kind had ten pulleys, and was constructed for the Tidal Committee of the British Association. A second machine on a larger scale, with twenty pulleys (and therefore giving the resultant of twenty S.H. components) has been constructed for the Indian Government and used for computing the tides at the principal Indian ports. Its pulleys do not travel in circles like those of the machine above described, but in vertical lines on the crank-and-slot principle of § 84, so that their motions are rigorously simple harmonic. The setting of the machine for the amplitudes and epochs of a given port occupies only a few minutes, and the tidal curve for a year can be drawn in four hours. Plate IV. is a representation, on about th of the original scale, of the tidal curve traced by this machine for the nineteen days commencing midnight August 1, and ending midnight, August 20, 1881, for Beypore in India. For the original from which it is reduced we are indebted to Mr. E. Roberts, of the Nautical Almanack Office, who has charge of the machine and its working. This method of combining a number of S.H. motions appears to have been first invented by the Rev. F. Bashforth, who printed and circulated a lithographed description of it in 1845, from which Fig. 30, in § 84, is taken. An abstract of Mr. Bashforth's paper will be found in the British Association Report' for that year (Transactions of Sections, pp. 3, 4). It was reinvented by Mr. W. H. L. Russell (see 'Philosophical Magazine,' 1870), and afterwards by Mr. Beauchamp Tower, who suggested it to Sir W. Thomson when in search of some convenient method for combining such motions with a view to the graphical prediction of tides. CHAPTER VIII. PROPAGATION OF SONOROUS UNDULATIONS. 97. THE propagation of sound depends upon the elasticity of the medium through which the sonorous undulations are propagated. For example, when a tuningfork vibrates in air, it gives the air a series of pushes, each of which produces a momentary increase of pressure and density in front of the advancing prongs, while a momentary decrease of density and pressure is produced behind them. As the prongs advance, first in one direction and then in the opposite, a series of compressions and extensions are produced in alternate succession. But each compressed portion tends to relieve itself by expanding into the neighbouring air, which is thus in its turn compressed, and the extended portions in like manner tend to communicate extension. Hence a series of compressions and extensions are propagated through the surrounding air, and these constitute an undulation, whose period is the same as that of the vibrations of the tuningfork. The velocity of propagation is independent of the period, and depends only on the elasticity and density of the air, being (as we shall prove in Chapter X.) directly as the square root of the coefficient of elasticity, and inversely as the square root of the density. 98. The coefficient of elasticity is to be understood in the following sense. Suppose a portion of air having the volume v to be slightly compressed so that its volume is Let P reduced to v―v, where is a small fraction. V denote the pressure per unit of area exerted by the air before and P+ after compression; then the quotient of p by - is called the coefficient of elasticity. V 99. The compression of air raises its temperature; and hence, if air is suddenly compressed, and then allowed to regain its original temperature, without further change of volume, its pressure immediately after compression is greater than that which it finally attains. But this final pressure is to its pressure before compression in the inverse ratio of the volumes, so that if this final pressure be p+p, we have |