E point . Imagine s' to be the X X/ 7 Let now Es' be represented by r; also let the co-ordinates EX', EY', S's of the spot s' be represented by x,y,z; then YES, the geocentric longitude of s', being represented by l, and s'Es, the geocentric latitude, by λ, we have, as in art. 184., x = r cos. A cos. l, y = r cos. A sin. l, and z = r sin. λ. Lastly, let the three rectangular co-ordinates of s', with respect to s, be x' (= sx") y' = (SY") and z = (s's), and let the semidiameter s s' of the sun be represented by r': then, X, and y'y Y; x' = x and in these equations substituting the above values, we have for the equivalents of a', y', and z, and subsequently of r', values identical with those in art. 184. From these, by processes similar to those in art. 185., we may obtain the tangent of the angle which the perpendicular let fall from s on the plane of the circle, described by the spot, makes with the axis sz"; that angle is equal to the inclination of the plane of the circle, and consequently of the sun's equator to the plane of the ecliptic. The required inclination is therefore found. α ß The position of the line in which the sun's equator intersects the ecliptic may be found by the method explained in art. 186.; in which the value of expresses the co-tangent of the angle which the line of section makes with sx", and being negative, the angle (as T'sq in the figure) is greater than one right angle and less than two: this angle represents the longitude of the node of the sun's equator. 286. It must be observed that the time in which the sun revolves on its axis is not precisely that which elapses between the first appearance of a spot on the eastern limb, and its reappearance at the same limb. For let E be the earth and ss' part of the sun's apparent annual path, so that ABC, A'B'C' may be considered as projections of the upper hemisphere of the sun on the plane of the ecliptic; and imagine the axis a C/ S B of the sun's rotation to be perpendicular to the plane of the ecliptic, which is sufficiently correct for the present purpose: also let a be the place where a spot becomes visible to a spectator on the earth, in a direction of a line drawn from E and touching the disk of the sun. Now while the spot A appears to revolve in the direction ABC, the sun appears to move in its orbit; therefore let the centre of the sun pass from s to s' while A describes about s the exact circumference of a circle: the spot will then be at a in such a situation that the angle as'B' is equal to ASB. But the sun must continue to revolve on its axis till the spot arrives at A', making the angle A's'E equal to ASE before it becomes visible; that is, it must describe the arc a A′ above a complete revolution about s. The arc a A', or the angle as'A' is equal to B'S'E or to s'Es, which in 27 days 8 hours is equal to 27°7'; consequently, the exact time of a revolution may now be found by proportion; thus E 387° 7′ 360° :: 27 days 8 hours: 25 days 10 hours, and the last term is the time required. 287. It may be seen by the naked eye that the surface of the moon is remarkably diversified with light and shade; and the telescope shows that it resembles the appearance which the earth would present to a spectator at a great distance from it if the vegetable mould which covers so great a part of the land were removed, and if the beds of the seas were dry. In many places great mountains rise from the general surface, and terminate in points, or form clusters or ridges; but much of the surface is occupied by deep cavities, which are surrounded by circular margins of elevated ground: there is no appearance of water in the moon, and the existence of an atmosphere about it is doubtful. 288. The spots on the moon's surface always retain the same positions relatively to each other, yet, apparently, they change their places with respect to the circumference of her disk in consequence of certain vibrations of the luminary on its centre, between the east and west, and between the north and south, which are called the librations in longitude and latitude, respectively. The first depends on the inequalities of the moon's motion in her orbit, combined with a uniform motion on her axis; in consequence of which motions the parts of her surface about the eastern and western limbs are turned alternately towards, and from the earth. The second depends on the axis of her rotation not being perpendicular to the plane of her orbit, and on its keeping parallel to itself during a revolution about the earth; thus a part of her northern or southern limb, which at one time may be unseen, is at another rendered visible. A third libration, called diurnal, results from a part of the moon's surface about the upper limb becoming visible when the moon rises and sets, the spectator being then above her, and ceasing to be seen when the moon has considerable elevation above the horizon, from the spectator being below her: at this time a part of the surface about the lower limb becomes visible, which in the other positions could not be seen. 289. With a micrometer applied to a telescope mounted equatorially, the difference between the right ascensions of a spot and of the eastern or western limb of the moon, also the difference between the declinations of a spot and of the upper or lower limb, may be found as the corresponding elements of the sun's spots were obtained; and from these observations the geocentric longitude and latitude of a spot, together with the values of a', y', z', may be determined. In employing the formulæ given in art. 184. there must be put for L the longitude of the moon, and for R its distance from the earth when projected on the plane of the ecliptic by means of the moon's latitude: the distance of a spot from the earth must be represented by r, its geocentric longitude by 1, and its geocentric latitude by λ. This last is equal to the sum or difference of the latitude of the moon's centre, and the distance of the spot in latitude from the moon's orbit. If the co-ordinates x, y, z, are determined on the supposition that s (fig. to art. 285.) is the centre of the moon projected on the plane of the ecliptic, the places of the moon's spots, as they would appear if observed from her centre, may be determined. For if s were the projection of a spot on the plane x' of the ecliptic, we should have equal to the tangent of its selenocentric longitude, and y' √x22 + y22 for the tangent of the corresponding latitude. Here must be considered as equal tor sin, A-R' sin. λ', R' being the distance of the moon from the earth and a' the moon's latitude. By means similar to those which are indicated in art. 185. the position of the moon's equator may be determined: and it is found that the latter forms nearly a constant angle (= 1° 30') with the plane of the ecliptic. The intersection of her equator, when pro duced, with the plane of the ecliptic, is found to be constantly parallel to the line of the nodes of her orbit. P E E n M m of light pro 290. The determination of the heights of the lunar mountains is an object rather of curiosity than of use in practical astronomy; but as it may be interesting to know in what manner those heights may be found, the following method, which was proposed by Sir William Herschel, is here introduced. Let c be the centre and AmN part of a section of the moon made by a plane passing through the earth and sun; let m be the summit of a mountain, and Smм the direction of a ray of light from the sun, touching the level part of the moon's surface at m, and enlightening the top of м while the space between м and m is in darkness; also let M E, mE' be the directions of rays ceeding to the earth, whose distance from the moon is great enough to allow those rays to be considered as parallel to one another. Then mp drawn perpendicularly to EM denotes the breadth of the unenlightened space Mm as seen from the earth, and this must be measured in seconds of a degree by means of the micrometer. Now the rays of light from the sun to the earth and moon being, on account of the great distance of the sun, considered as parallel to one another; if sм be produced to meet CP, which is drawn parallel to ME and is supposed to join the centres of the earth and moon, in the point P, and PY be a line drawn to the equinoctial point in the heavens, the angle CPS or its equal pмm will be equal to the difference between the longitudes of the sun and moon (YPS and TPC) at the time of the observation: this angle may be found from the Nautical Almanac; and in the plane triangle мpm right angled at p, sin. mмp: rad.:: mp Mm. Thus Mm may be found in seconds of a degree; but the angle subtended at the earth by the semidiameter of the moon, or the value of cm, is known from the Nautical Almanac; therefore in the triangle CMm right angled at m, CM may be found, and subtracting cm or cn from it, we have Mn, the height of the mountain. This is expressed in seconds; but since the semidiameter of the moon is known in miles, the value of мn may, by proportion, be found in miles. From the computations of Sir William Herschel it appears that the heights of the mountains above the level surface of the moon do not exceed 11⁄2 miles. CHAP. XIII. THE FIXED STARS. REDUCTION OF THE MEAN TO THE APPARENT PLACES. -THEIR PROPER MOTIONS. ANNUAL PARALLAXES. 291. AN exact knowledge of the positions of the stars called fixed is of the highest importance since on those positions depend the determination of the places of the sun, moon, planets, and comets; and consequently the verification of the results of theory respecting the movements of the bodies composing the solar system. It has been shown that the apparent places of stars are affected by the causes which produce the precession of the equinoxes, the solar and lunar nutation, and what is called the aberration of light; and it may be proper to state here the formula by which, for any given time, the apparent places with respect to right ascension and declination may be reduced to the mean places, and the converse. In the present Nautical Almanacs (page 435.) these formulæ are indicated by the two following series, the values of the several terms being taken from Mr. Baily's paper, "On some new Tables," &c., in the Memoirs of the Astronomical Society, vol. ii. part 1.: (in} The apparent right ascension (in ) seconds of a degree) The apparent declination =a+Aa + Bb + Cc + Dd. =d+Aa'+Bb'+cc' + Dd. In these formulæ a and 8 are the mean right ascension and mean declination at the commencement of the year, as in the tables, pp. 432 to 434. of the Almanac: the terms indicated by AaBb constitute the formulæ for aberration in right ascension (art. 234.), and those indicated by Aa' + Bb' constitute the formulæ for aberration in declination (art. 235.). The terms Cc Dd comprehend all those which enter into the effects of luni-solar nutation in right ascension, and Cc Dd' all those which enter into the effects of luni-solar nutation in declination. The quantities represented by A, B, C, D, depend on the true longitudes of the sun and moon and the longitude of the moon's node; and as these vary continually their logarithms are given for every day in the year, in the Nautical Almanacs, page xxii. of each month. A further |