ON MAPPING THE SURFACE OF THE MOON. 227 



15. To calculate the librations of the centre of the apparent disk, it will be 

 necessary, fii-st, to determine the selenocentric coordinates of the point or, as 

 referred to the great circle 7" N, c, fig. 2, parallel to the plane of the ecliptic. 



In fig. 5 let the angle at ^^^=1, the inclination of the moon's equator 

 to the ecliptic, N^ m, as before, fig. 2, representing an arc of the moon's 



rig. 5. 



A a 



equator, and N^ C (c fig. 2) an arc of the echptic. As the arc N^ M is the pro- 

 jection 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 geo- 

 centric longitude of the moon and the longitude of the ascending node of the 

 moon's equator ?3, Let .•. X be the moon's geocentric longitude, then 

 N^ M=\— ?g . Let A' be an arc measured from T to N^, fig. 2, and then 

 from N^ to L", fig. 5, so that N"^ L"=A'- ?g , and L"jp = Z-A' { •.• I is also 

 measured from 7"}. 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'— JS to B' ; then by the right- 

 angled spherical triangle L" N^ M we have 



tan (A'— ?S) = tan(X— ?S), secl, 



tan B' =sin (\ — ?3 ) tan I, 



cos 0=cos (\— ?8 ) sin l=a', in the ' Nautical Almanac,' 



COS I 



sin 0= — ,, and by the right-angled spherical triangle 



COS Jj 



L" crp, putting /3, the geocentric latitude of the centre, for <r M, 



tan (Z— A')=cos tan (/5— B'), 

 sin ^j=sin 6 sin (/3 — B'). 



Formulce for Lihration in Latitude. 



16. Libration in latitude, or the selenographical latitude of the centre of 

 the apparent disk, is equal to the angle subtended between the point o-, the 

 centre of the apparent disk, and the point p the abscissa on the moon's 

 equator, to which it is referred, so that a- p 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 — 6'=^j (see section 9), it follows that 6'=B'— ^, for 0j, or ft—'S! 

 (i. e. a-p), is the Ubration in latitude apart from its sign. As ^^ is positive 

 when the point tr,^ (see section 9) is above the moon's equator (for which 

 Z,=/\ nearly), it will in the same case be negative for the point or (see section 

 9) (for which Zj=\ + 180° nearly), but in the case supposed the libration 

 in latitude is negative ; hence if V= this libration, 6'= — 0,=B'— /3, which 

 is the expression in the ' Nautical Almanac' 



q2 



