ON MAPPING THE SURFACE OF THE MOON. 233 



from mean libration occurs, the real path of the intersecting point on the 

 apparent disk is a wide ellipse, which gradually contracts to a narrow ellipse 

 as the epoch of mean libration is approached. This will be the case propor- 

 tionally with every point on the apparent disk, and the displacement will be 

 in every possible direction and at every conceivable angle with the centre of 

 the apparent disk. This suggests that by far the most effective mode of de- 

 termining positions on the moon's surface is by measures for points of the 

 first order ; for let x' y' represent the measures in right ascension and decli- 

 nation from the east and north limbs of the point E, x and y wiU be the 

 corresponding rectangular coordinates necessary to determine the selenogra- 

 phical position when the librations of the centre and the other elements are 

 ascertained. 



Computation of Points of the Fikst Order. 

 Measures. 



In order to compute the selenographical coordinates of a point on the moon's 

 surface (its latitude and longitude), the following measures are necessary : — 



Between five and ten measures of the distance of the point from the illu- 

 minated north or south limbs ; also from the Uluminated east or west limbs as 

 the case may be. 



The moon's diameter in the direction of the line drawn through both cusps, 

 which may be assumed perpendicular to the ecliptic as the moon seen from 

 the sun, departs at the utmost only 50" from the plane of the ecliptic. 



These measures require to be corrected for refraction according to the fol- 

 lowing formulae. 



The measured diameter in micrometer revolutions, which call D, is to be 



multiplied by the factor ( l-|-^^^-^iii J, in which w'= the angle which the 



line of the cusps makes with the vertical circle passing through the moon, 

 dr= the difference of refraction in seconds for 10' in the altitude of the moon 

 (within the narrow space of the moon's disk the difference of refraction may be 

 assumed proportional to the difference of altitude) ; dr may be taken from the 

 ' Connaissance des Temps.' 



The formulae for correcting the measured distances (also in micrometer 

 revolutions) of the point from the moon's limbs are as follows : — 

 In the declination circle, 



AA . dr . cos n 

 25-12x10 ' 

 in the parallel, 



A^' . dr . sin n _ 

 25-12x10 ' 



n representing the inclination of the apparent declination circle to the ver- 

 tical circle passing through moon's centre, and A/i, A/i' the differences of alti- 

 tude of the measured point and the tangents at the respective limbs. When 

 southerly the correction is + for heights and — for depths, and the reverse 

 when northerly. 



Having obtained these measures and corrected them, the following elements 

 for the time of observation should be taken from the ' Nautical Almanac' 



Elements. 

 a. = the true right ascension of the moon. 

 6 = the true declination of the moon. 



