224 - REPORT— 1866. 



is the great circle parallel to tlie earth's equator, v its pole, Y'N'^c is tho 

 great circle parallel to the plane of the ecliptic. 



5. In the spherical triangle T N^ N^ the angle N'j 7" N'^ = w the obliquity 

 of the ecliptic, the angle y N, Nj=I, the inclination of the moon's equator 

 to the ecliptic, and X'^^—'^y the longitude of the ascending node of the 

 moon's equator. 



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, 300° must be subtracted. 



G. Let the angle N.^ N, e=i, the inclination of the moon's equator to the 

 earth's equator, ^yhich is equal to the arc m e or P tt. 



7. Let T 1^1= S'> the distance fi-om 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 N2=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 formulas in spherical trigonometry, 



^ cos|(w-I)^ ?3 ^ ^ sin^(a,— I) 83 



tan A=— — T7 — TT\ t'™ "o') ^^^^ li=^-7— 1 / , tn tan -y 

 cosg(w + i) .:: sm 3 (w + i; z, 



. i sin Hw— I) . ?3, ^ a , t) ^ - a -n 



sm —= ;=^3 sin-^' A = A + B, £3 =A— B. 



2 sm B 2 



These are the formuloe for calculating the values of i A and S ' given on 

 p. X of the ' Nautical Almanac' 



8. Let a, fig. 2, be any point in the celestial sphere, of which the posi- 

 tion is given by the selenocentric longitude 1, reckoned from T to N^, and 

 then along the moon's equator to p, and by the selenocentric latitude, 

 ffp=(j,^. Also, let the coordinates of the same point referred to the plane 

 parallel to the earth's equator be T ra- = a,, and w a=S^, and let the angle 

 P(r7r = C'; then in the triangle P tt (t, P Tr=:i, P <7 = 90°-(/,„ tt (r = 90°-S„ 

 the angle tt P (r=90°- N, ^^ = 90°- (N^^^ + N^N.,) = 90°- (1,- ?S +A) 

 {•.• r N,= ?8 }, and the angle Ptt (r=180°-w f =180°-(90°-N, ^!r)=90° 

 4-N,Br=90°+ai-g'; {•.• rN^=83'}. Hence 



sin C _ sin (90° - (1, - gg + A) ) ! cos (1, - gg + A) 

 sin i ■ sin (90°-^J ~ cos S, ' ^^' 



sin C sin(90° + g,— £3') _cos(g,- £3') 



sin i ~ sin (90°— ^J cos cp^ 



9. The equations in section 8 relative to the angle C arc general, the 

 point a- not having been defined. If, however, we suppose that <r and a-^ 

 represent those points on the moon's surface that are cut by the line joining 

 the centres of the earth and moon, and that <r is situated on the hemisphere 

 turned towards the earth, <r^ wiU be situated on the opposite or superior 

 hemisphere. Now i^;=the selenocentric latitude of the point <r, and this is 

 equal to the selenocentric latitude of the centre of the apparent disk, as o- is 

 in the line joining the centres of the earth and moon, but this is equal to 

 the geocentric latitude ivlth opposite slr/ns, i.e. if i'=the moon's geocentric 

 latitude ^1 = — V. 



In adapting the formulce in section 8 to the position of tr, viz. towards 

 the earth, let a' and h' be the moon's apparent M, and N.P.D. corrected for 

 parallax, then by the definition of the point ir, a^ =180° + a', 0, = — ?>', 

 %^= —l', and 1^=1+ 180°, Idiifering from \, the moon's geocentric longitude 



