Through the 24 partial sums or means finally obtained for a given component (and its har monics) imagine a curve to be drawn. It may be represented by a Fourier series whose coef ficients and initial phases can be found without difficulty. Then by taking the initial phases from the proper astronomical arguments for that time, the required epochs will be obtained.
14. Prediction of At most ports the time of tide can be roughly predicted by adding to the time of the moon's transit, upper or lower, the average value of the lunitidal inter val or "establishment" for the port. Where the diurnal wave is not large, the height of high water above mean sea-level is roughly equal to half of the mean range of tide; the low water is as much below this plane or datum.
Where great refinement is desired, the proc ess just referred to involves much labor, and can best be carried out after a tolerably com plete harmonic analysis has been made. This done we have only to add together a series of cosine terms for obtaining the height of the surface of the sea at any given time. But the number of these terms is so great that such a computation would soon become laborious, since it must be made for many assumed times in each day's predictions. Several predicting in struments have been designed or constructed. Generally speaking, the object of such an in strument or machine is to produce simple har monic motions of suitable periods, phase and amplitude and to combine the results into one compound wave, or perhaps into two com pound waves. The simplest form may be described thus: Upon one or more shafts, driven by hand or by clockwork, are fixed a number of wheels which mesh into other wheels, causing the latter (or wheels moved by them) to revolve with angular velocities hav ing given ratios to the angular velocities of the shafts. These ratios are taken, as nearly as possible, proportional to the speeds of particular tidal components. Rigidly connected to these wheels are cranks carrying pulleys, or pins working in slots and which impart to vertical rods carrying pulleys rectilineal harmonic mo tions. At one end of the machine a chain or flexible wire is made fast ; thence it is laid alter nately over and under the pulleys. Near the
other end of the chain or wire is attached a marking point which, when moved transversely to the line of motion of the paper roll, traces the tidal curve. It is evident that the machine continually sums a series of cosine terms, If the instrument include a set of cranks at right angles to those giving the tide curve, to gether with pulleys and a summation chain or wire, the movement of a point on the chain (taking the place of the tracing point) across a fixed line or mark, corresponding to mean sea level, will show the time when a high or low water occurs.
The original and simplest form of instru ment is that due to Kelvin. The one designed by Ferrel is somewhat complicated, inasmuch as it separates out the principal lunar component. It was used in the office of the United States Coast and Geodetic Survey for about 30 years. Its chief advantage consists in indicating upon its face the times and heights of the tides. The office has recently completed and put in opera tion a machine which besides tracing a curve, as does Kelvin's indicates upon its face the times and heights of the tides,— the times be coming known through the additional mechan ism just described.
Bibliography.— Newton, (Bk. I, Prop. 66; Bk. III, Props. 24, 36 and 67) ; La place, (Traite de Mecanique Celeste) (Bks. IV and '(III) ; Sir John W. Lubbock, Rev. Wm. Whewell, and others in the 'Philosophical Transactions' (1830 et seq.) ; Airy, (Tides and Waves) (Enc. Met.) ; Thomson and Tait, ural Philosophy' (2d ed., §§ 804-848 and Appen dix B) ; Ferrel, 'Tidal Researches' (United State Coast Survey) ; Lamb, (Hydrodynamics' ; Darwin, (The Tides and Kindred Phenomena in the Solar "Scientific Papers"; Maurice Levy, (Lecons sur la Theorie des Maress); H. Poincare (Lev:ins de Mecanique Celeste' ; United States Coast and Geodetic Survey Reports; Ap pendixes by A. D. Bache, Chas. A. Schott, Henry Mitchell, Wm. Ferrel, R. A. Harris and others.
Annual.— Tide Tables issued by the govern ments of Great Britain, France, Germany, the United States and other countries.