Assuming that the day has not altered in length since the earth was in a fluid state, or when we have observed ellipticity of 1 k 1 ellipticity due directly to diurnal rotation. That is, 1 1 1 '578' whence ke----.0.486.1 The value of 1/8 can be determined by com paring the observed period of the movement of the axis of figure about the axis of rotation with the period of free precession (or 305 sidereal days) obtained upon the assumption of a rigid earth.
Another determination of ke is obtained by comparing the observed range of the monthly or fortnightly tide with the equilibrium range due to the direct action of the tidal forces. This ratio or m/a is found to be about M. Daily or semi-daily tides in a deep lake or in a buried pipe also give a determination of m/a. The actual deviation of the vertical from its mean position, due to the daily or semi-daily tidal forces, and obtained by means of a hori zontal pendulum, gives, upon comparison with the direct theoretical deviation, another de• termination of the ratio m/a. If kc he taken as 0.5, and m as M, then by the above equation, =0.5 and a; that is, the actual rise and fall of the surface of the "solid* earth is about 35 of the rise and fall of the surface of an ocean of small density surrounding a rigid sphere. If the latter be 1.8 feet (§ 5), the former will be about 1.2 feet. [This mode of treatment is due to Ch. Lallemand] 12. Observation of The height of the tide, or water surface, at any given time can be directly observed upon a graduated fixed staff if the water be tolerably smooth. A long box fixed in a vertical position, having a small opening near the lower end, and usually sup plied with a float, enables one to observe during stormy weather as well. In some instances only the times and heights of the high and low waters are observed and recorded. But a more satisfactory record is obtained by reading the gauge at regular intervals one hour or one-half hour apart.
Automatic or self-registering gauges are usually constructed for drawing a continuous curve. Such a gauge consists of a float and box, a time-piece, and some form of recording apparatus. The record from an automatic gauge can be procured with comparatively little trouble; it shows clearly the peculiarities of the tide; and it furnishes material for a thorough analysis or discussion.
The velocities and directions of tidal streams are usually ascertained by aid of a float and line or some form of current-metre. Such observa tions are attended with considerable difficulty because they must generally be made at some distance off shore.
13. Analysis of If only the high and low waters are to be treated, they are first referred to the moon's transit for obtain ing lunitidal intervals. The ranges of tide are found at the same time. The intervals and the heights or the. ranges are next classified accord ing to certain astronomical arguments for the purpose of bringing out the corresponding tidal inequalities.
The harmonic analysis rests upon Laplace's principle of forced oscillations, namely: The state of any system of bodies, in which the primitive conditions of the motion have dis appeared through the resistances which the mo tion encounters. is coperiodic with the forces acting on the system.
Here is the clue to what oscillations ought to be found in the tidal wave; for, there ought to be an oscillation corresponding to each term of the causes producing the tide. Such terms follow from the development of the tide pro ducing potentials of the moon and sun. Their arguments and "speeds" involve simple com binations of the mean longitudes and mean mo tions of the local meridian projected on the celestial sphere, the moon, the sun, the equinox, the lunar perigee, the solar perigee, and the moon's node. There are three principal classes of terms: semi-diurnal, diurnal and long-period.
If for a sufficiently long time the observation curve be read and summed with reference to any constituent or component defined by its "speed," the effects of the other components will gradually disappear and the final sums will pertain only to the one sought (including, of course, its harmonics). To avoid reading the curve with reference to each component, the tabulation according to mean solar time is made to serve for all. This is done by dis tributing the (solar) hourly heights among the component hours as nearly as possible. Tables showing the proper distribution of /lowly heights as well as blank forms into which the heights are to be copied have been used quite extensively. For about 30 years perforated sheets, known as stencils, have been used for pointing out which hourly heights of the tabula tion go with the various hours of the com ponent sought. These enable one to dispense with the labor involved in copying into the various forms. Some years ago a set of mov able scales or strips was devised upon which the hourly heights are copied once for all sum mations. Several machines have been devised or constructed for facilitating the work of analysis, but they have not vet come into actual use.