mutual The tides, the meridian of Greenwich, of which the five constant coattraction of efficients depend merely on the configuration of land and water, neglected: and may be easily estimated by necessarily very laborious equilibrium quadratures, with data derived from the inspection of good the waters corrected theory. maps. Let as above r = a (1 + u)... .....(14) be the spheroidal level that would bound the water were the whole solid covered; u being given by (13) of § 804. Thus, if ffdo denote surface integration over the whole surface of the sea, affudo expresses the addition (positive or negative as the case may be) to the volume required to let the water stand to this level everywhere. To do away with this change of volume we must suppose the whole surface lowered equally all over by such an amount a (positive or negative) as shall equalize it. Hence if be the whole area of sea, we have is the corrected equation of the level spheroidal surface of the sea. Hence where h denotes the height of the surface of the sea at any place, above the level which it would take if the moon were removed. To work out (15), put first, for brevity, Now let 7 and λ be the geographical latitude and west longitude of the place, to which u corresponds; and and 8 the moon's hour-angle from the meridian of Greenwich, and her declination. As is the moon's zenith distance at the place (corrected for parallax), we have by spherical trigonometry which gives 3cos 0-1=cos2lcos2dcos2(x − y)+6sinlcos/sindcosdcos(λ − y) + †(3sin2♪ - 1) (3sin2 - 1) (20). Hence if we take A, B, C, D, E to denote five integrals depend- The tides, ing solely on the distribution of land and water, expressed as attraction of mutual the waters neglected: corrected equilibrium theory. a= = {[fudo =}ar{}cos2d(Acos24+Bsin24) + 6sindcos♪(Ccosy + Dsiny) + {E(3sin2♪ — 1) } (22). This, used with (19) and (20) in (17), gives for the full conclu sion of the equilibrium theory, h = ar [(cos3l cos 2λ – A) cos 24 + (cos3 l sin 2λ – B) sin 24] cos2 8 + }ar (3 sin2 l − 1 − C) (3 sin2 8 − 1) in which the value of 7 may be taken from (18) for either the 809. In the equilibrium theory, the whole deviation of level at any point of the sea, due to sun and moon acting jointly, is expressed by the sum of six terms, three for each body. solar semi (1) The lunar or solar semi-diurnal tide rises and falls in Lunar or proportion to a simple harmonic function of the hour-angle from diurnal tide. the meridian of Greenwich, having for period 180° of this angle (or in time, half the period of revolution relatively to the earth), with amplitude varying in simple proportion to the square of the cosine of the declination of the sun or moon, as the case may be, and therefore varying but slowly, and through but a small entire range. Lunar or solar di (2) The lunar or solar diurnal tide varies as a simple harurnal tide. monic function of the hour-angle of period 360°, or twenty-four hours, with an amplitude varying always in simple proportion to the sine of twice the declination of the disturbing body, and therefore changing from positive maximum to negative, and back to positive maximum again, in the tropical* period of either body in its orbit. Lunar fortnightly tide or solar semi-annual tide. Explanation of the lunar fort nightly and solar semi annual tides. (3) The lunar fortnightly or solar semi-annual tide is a variation on the average height of water for the twenty-four lunar or the twenty-four solar hours, according to which there is on the whole higher water all round the equator and lower water at the poles, when the declination of the disturbing body is zero, than when it has any other value, whether north or south; and maximum height of water at the poles and lowest at the equator, when the declination has a maximum, whether north or south. Gauss's way of stating the circumstances on which "secular" variations in the elements of the solar system depend is convenient for explaining this component of the tides. Let the two parallel circles of the north and south declination of the moon and anti-moon at any time be drawn on a geocentric spherical surface of radius equal to the moon's distance, and let the moon's mass be divided into two halves. and distributed over them. As these circles of matter gradually vary each fortnight from the equator to maximum declination and back, the tide produced will be solely and exactly the "fortnightly tide." 810. In the equilibrium theory as ordinarily stated there is, at any place, high water of the semi-diurnal tide, precisely when the disturbing body, or its opposite, crosses the meridian of the place; and its amount is the same for all places in the same latitude; being as the square of the cosine of the latitude, and therefore, for instance, zero at each pole. In the corrected *The tropical period is the interval of time between two successive passages of the tide-raising body through the intersection of the orbit of that body with the earth's equator. In the case of the moon this intersection oscillates, with a period of 18 years, through about 13o on each side of the first point of Aries, as the nodes of the lunar orbit regrede on the ecliptic (see § 818 a, b). In the case of the sun the intersection is the first point of Aries, which completes its revolution in 26,000 years. mutual at the waters practical of correction brium fort semi-an equilibrium theory, high water of the semi-diurnal tides may The tides, be either before or after the disturbing body crosses the meri- traction of dian, and its amount is very different at different places in the neglected: same latitude, and is certainly not zero at the poles. In the importance ordinarily stated equilibrium theory, there is, precisely at the for equilitime of transit, high water or low water of diurnal tides in nightly and the northern hemisphere, according as the declination of the nual tides. body is north or south; and the amount of the rise and fall is in simple proportion to the sine of twice the latitude, and therefore vanishes both at the equator and at the poles. In the corrected equilibrium theory, the time of high water may be considerably either before or after the time of transit, and its amount is very different for different places in the same latitude, and certainly not zero at either equator or poles. In the ordinary statement there is no lunar fortnightly or solar semi-annual tide in the latitude 35° 16′ (being sin1 1/3), . and its amount in other latitudes is in proportion to the deviations of the squares of their sines from the value. In the corrected equilibrium theory each of these tides is still the same in the same latitude, and vanishes at a certain latitude, and in any other latitudes is in simple proportion to the deviation of the squares of their sines from the square of the sine of that latitude. But the latitude where there is no tide of this Latitude of evanescent tide. class is not sin(1/√/3), but sin ̄1[√/ (1 + E)], where is the fortnightly mean value of 3 sin'l-1, for the whole covered portion of the earth's surface. In § 848 c below will be found an approximate evaluation by means of quadratures of the function E, contributed by Mr G. H. Darwin to our present edition. The uncertainty as to the amount of land in arctic and antarctic regions renders this evaluation to some degree uncertain; but it appears in any case that the distribution of the land is such that the latitude of evanescent fortnightly tide is only removed a little to the southward of 35° 16'. The computations show, in fact, that this latitude is 34°40′ or 34°57', according to the assumptions made as to the amount of polar land. As the fortnightly and semi-annual tides have been supposed by Laplace to follow in reality very nearly the equilibrium * * In our first edition we undoubtingly accepted this supposition. Spring and neap tides: and "lagg. law, the determination of the latitude of evanescent tide is a matter of great importance. It is moreover possible that careful determination of the fortnightly and semi-annual tides at various places, by proper reductions of tidal observations, may contribute to geographical knowledge as to the amount of water-surface in the hitherto unexplored districts of the arctic and antarctic regions. 811. The superposition of the solar semi-diurnal on the "priming" lunar semi-diurnal tide has been investigated above as an ing." example of the composition of simple harmonic motions; and the well-known phenomena of the "spring-tides" and "neaptides," and of the "priming" and "lagging" have been explained (§ 60). We have now only to add that observation proves the proportionate difference between the heights of Discrepancy spring-tides and neap-tides, and the amount of the priming served re- and lagging to be much less in nearly all places than estimated in § 60 on the equilibrium hypothesis; and to be very different in different places, as we shall see in Vol. II. is to be expected from the kinetic theory. from ob sults, due to inertia of water. Lunar and solar in apparent gravity. 812. The potential expressions used in the preceding investigation are immediately applicable (§§ 802, 804) to the hydrostatic problem. But it is interesting, in connexion with this problem, to know the amount of the disturbing influence on apparent terrestrial gravity at any point of the earth's surface, produced by the lunar or solar influence. We shall therefore fluence on still using the convenient static hypothesis of § 804—deterterrestrial mine convenient rectangular components for the resultant of the two approximately equal and approximately opposed disturbing forces assumed in that hypothesis. First, we may remark that these two forces are approximately equivalent to a force equal to their difference in a line parallel to that of the centres of the earth and moon, compounded with another perpendicular to this and equal to twice either, multiplied into the cosine of half the obtuse angle between them. Resolving each of these components along and perpendicular to the earth's radius through the place, we obtain, by a process, the details of which we leave to the student, the following results, which are stated in gravitation measure:- |