larger scale, in each of which there should be a point of the first orderbeing apparent, this Appendix contains a form of computation based on Encke's method, and modified from Lohrmann's and Beer and Mädler's works, and as libration enters as a necessary element into the calculation, it is preceded by an investigation of libration in latitude and longitude. For the MS. from which the greatest part of this investigation is taken I am indebted to A. C. Ranyard, Esq., of Cambridge. I must, however, remark that the formulæ are derived from the 'Berliner Astronomisches Jahrbuch für 1843.' 2. The investigation of libration consists of three parts, viz., that of the angle C, or the angle which the meridian passing through the middle of the moon's apparent disk makes with the circle of declination; that of libration in latitude and that of libration in longitude. The meridian passing through the middle of the apparent disk should be carefully distinguished from the first meridian on the moon's surface, from which all selenographical longitudes are reckoned both east and west. 3. It will greatly assist in the conception of libration if the following principles be borne in mind. Fig. 1. Three planes being supposed to pass through the moon's centre, viz. the plane of the moon's equator, the plane of her orbit, and a plane parallel to the plane of the ecliptic, the last will lie between the others, and will intersect them in the line in which they intersect each other. In consequence of this law the longitude of the ascending node of the moon's equator on the ecliptic always differs by 180° from the longitude of 5.9' U Ecliptic 1° 32.9" b Moon's Equator the ascending node of the orbit. The inclination of the moon's equator to the ecliptic is 1° 32' 9", the inclination of the plane of the orbit is about 5° 9'. Investigation of the angle C. 4. Conceive the moon's centre to be the centre of the celestial sphere. In Earths Equator fig. 2, l' is the pole of the moon's equator, N, p m the moon's equator, Y N ̧ e is the great circle parallel to the earth's equator, its pole, r N, c is the great circle parallel to the plane of the ecliptic. 1 2 = 5. In the spherical triangle YN, N, the angle N, YN,w the obliquity of the ecliptic, the angle Y N, N1I, the inclination of the moon's equator to the ecliptic, and N, 8, the longitude of the ascending node of the moon's equator. = 2 This last quantity is obtained by adding 180° to the longitude of the ascending node of the orbit given in p. 242 of the Nautical Almanac.' If the sum exceed a whole circumference, 360° must be subtracted. 6. Let the angle N, N, e=i, the inclination of the moon's equator to the earth's equator, which is equal to the arc me or Pπ. 7. Let N, 8', the distance from the first point of Aries of the ascending node of the moon's equator on the earth's equator, or the right ascension of the ascending node of the moon's equator on the earth's equator, and N, N=A, the arc between the two nodes on the moon's equator, or the arc on the moon's equator from its ascending node on the earth's equator to its ascending node on the ecliptic. Then, by known formulæ in spherical trigonometry, These are the formula for calculating the values of i ▲ and ' given on p. x of the Nautical Aimanac.' 8. Let o, fig. 2, be any point in the celestial sphere, of which the position is given by the selenocentric longitude 1, reckoned from Y to N2, and then along the moon's equator to p, and by the selenocentric latitude, σp=1. Also, let the coordinates of the same point referred to the plane parallel to the earth's equator be Ya,, and wo=d,, and let the angle P=C'; then in the triangle Pπσ, Pπ=i, Po=90°-41, πσ=90°—d1, the angle Po=90°— N, p=90°- (N2p+N, N1) = 90°— (1,— 8+▲) {r N, 8}, and the angle Pσ=180-we=180°-(90°-Ń, )=90° +N1 =90°+a1-8'; {·.·• r N, = 8'}. Hence T= 9. The equations in section 8 relative to the angle C' are general, the point not having been defined. If, however, we suppose that and σ, represent those points on the moon's surface that are cut by the line joining the centres of the earth and moon, and that is situated on the hemisphere turned towards the earth, σ, will be situated on the opposite or superior hemisphere. Now the selenocentric latitude of the point σ, and this is equal to the selenocentric latitude of the centre of the apparent disk, as σ is in the line joining the centres of the earth and moon, but this is equal to the geocentric latitude with opposite signs, i. e. if b' the moon's geocentric latitude -b'. = In adapting the formulæ in section 8 to the position of σ, viz. towards the earth, let a' and ' be the moon's apparent AR and N.P.D. corrected for parallax, then by the definition of the point a, a, =180°+a', $1 = −b', dd', and 1=1+180°, 1 differing from A, the moon's geocentric longitude = by a small angle found subsequently. These substitutions being made in the formulæ for sin C', and changing C' into C, we have which are the formulæ on p. x of the 'Nautical Almanae' for computing the angle C. 10. The angle C changes sign with cos (a'-8') and i, the change of sign of i being due to the motion of the moon's nodes. It does not change sign with the changes of sign of d' and b'. It is positive when the northern part of the circle of declination is to the west of the moon's meridian. 11. In fig. 3, from Lohrmann, we have MP, the moon's pole; EP the earth's pole; M the centre of the apparent disk; (M P) (E P)=i the inclination of the moon's equator to the earth's equator; (EP) M=p', the N.P.D. of the moon's apparent centre 90°+&'; (MP) M=a, the distance of the moon's apparent centre from the moon's pole=P σ, fig. 2, =90°—91; the angle (MP) (EP) M-A the inclination of i to p'=90°+a'-8' (see section 8), or 270°+8'-a' (see Lohrmann, 'Topographie der Sichtbaren Mondoberfläche,' p. 27); the angle (E P) (MP) M=B the inclination of i to a=90°— (-8+4) (see section 8); the angle (EP) M (M P)=C the inclination of p' to a (see section 8, angle Poπ=C'). The formulæ for computing the angle A and the sides i and p' are given above. The Gaussian formulæ for obtaining the values of B and C, with the side a are as follows: B= (B+C)+(B-C) C=1 (B+C)— (B-C), (1) (2) (3) (4) These formulæ are employed in the following computations for determining the angle C and points of the first order. Investigation of Libration. 12. In mapping the surface of the moon the orthographical projection is used in which the centre is characterized by 0° of latitude and 0° of longitude. This point, of course, is that in which the moon's equator and first meridian intersect each other. We have consequently to deal with two points, or the centre of the apparent disk, which is the only point recognized in the computations of libration, and the point of intersection of the first meridian and the equator. These points coincide only when the line joining the centres of the earth and moon passes through the centre of the apparent disk in mean libration, which occurs in periods of 2.997 years. 13. At any other epoch than that of mean libration the point is removed more or less from the point of intersection of the equator and first meridian, consequently as is the only point of the moon's surface turned towards the earth to which the computations of libration refer, libration in latitude=the selenographical latitude of the apparent centre, and libration in longitude= the selenographical longitude of the same point. 14. When the moon passes the ascending node as seen from the centre of the earth, the moon's equator appears as a straight line on the apparent disk, and may be thus represented on the orthographical projection. Libration in latitude then=0°. As the moon passes from the ascending node to the greatest north latitude, the southern parts of the moon come into view, and the equator is projected on the apparent disk as the lower segment of a narrow ellipse, as given in an inverting telescope. All the appearances described in this Appendix are inverted, lower for upper, &c. The east limb or margin of the moon is seen in the telescope opposite to the right hand. The greatest libration in latitude = the moon's latitude + the inclination of the moon's equator to the ecliptic a b, fig. 1, p. 223. Were the moon a transparent globe and the equator marked on it, the equator would be seen as a long, narrow ellipse, widening and closing up between the passages of the nodes, so that at the passage of the descending node the libration is again = 0°. The same phenomena take place as the moon describes the portion of her orbit south of the plane of the ecliptic, but in the opposite sense, the northern parts coming into view. From this it will be seen that libration in latitude changes its sign every lunation at the passages of the nodes. 15. To calculate the librations of the centre of the apparent disk, it will be necessary, first, to determine the selenocentric coordinates of the point σ, as referred to the great circle Y N, c, fig. 2, parallel to the plane of the ecliptic. In fig. 5 let the angle at N=I, the inclination of the moon's equator to the ecliptic, N, m, as before, fig. 2, representing an arc of the moon's .. M C 2 equator, and N, C (c fig. 2) an arc of the ecliptic. As the arc N, M is the projection on the ecliptic of the arc subtending the angle at the moon's centre, contained between a line parallel to the nodal line and the line joining the centres of the earth and moon, it must be equal to the difference of the geocentric longitude of the moon and the longitude of the ascending node of the moon's equator 8. Let A be the moon's geocentric longitude, then N2 M=λ— 8. Let A' be an arc measured from Y to N,, fig. 2, and then from N, to L", fig. 5, so that N, L"A'- 8, and L" p=1-A { is also measured from r}. Also let L" M=B', the arc subtended between the moon's equator and the ecliptic, of which the greatest value=1° 32′ 1′′, and the angle N, L" M=0, the inclination of A'-8 to B'; then by the rightangled spherical triangle L" N, M we have 2 tan B' sin (A- 8) tan I, cos 0=cos (8) sin I=a', in the Nautical Almanac,' sin 0= cos I cos B" and by the right-angled spherical triangle L" p, putting ß, the geocentric latitude of the centre, for a M, σ 16. Libration in latitude, or the selenographical latitude of the centre of the apparent disk, is equal to the angle subtended between the point, the centre of the apparent disk, and the point p the abscissa on the moon's equator, to which it is referred, so that op is equal to the perpendicular dropped from the centre of the apparent disk upon the moon's equator. This angle is equal to the distance of the moon's apparent centre from the moon's pole, minus 90°, and is consequently equal to 0° when the moon is in either node. 17. As b'= (see section 9), it follows that b'B'-6, for 4,, or ß-B' (i. e. σp), is the libration in latitude apart from its sign. As o, is positive when the point σ, (see section 9) is above the moon's equator (for which =λ nearly), it will in the same case be negative for the point (see section 9) (for which =λ+180° nearly), but in the case supposed the libration in latitude is negative; hence if this libration, b'= —4,—B'—ß3, which is the expression in the Nautical Almanac.' |