210 PROCEEDINGS OF THE AMERICAN ACADEMY 



New Method for facilitating the Conversion of Longitudes and Lati- 

 tudes of Heavenly Bodies, near the Ecliptic, into Right Ascensions 

 and Declinations, and vice versa. 



In the computation of a Lunar Ephemeris, the conversion of the lon- 

 gitudes and latitudes into right ascensions and declinations forms no 

 inconsiderable part of the work to be done. Prof. Hansen, at the end 

 of his " Tables de la Lune," has given some tables, with the view of 

 diminishing the amount of labor required in this conversion. 



But their employment seems to me to possess little, if any, advantage 

 over the use of the ordinary formula? of spherical trigonometry. I pro- 

 pose the following method, which perhaps in a slight degree is more 

 ready than that of Prof. Hansen. 



Designating the right ascension, declination, longitude, latitude, and 

 the obliquity of the ecliptic respectively by a, 8, I, b and e, we have the 

 following equations 



sin S = cos e sin b -f- sin e cos b sin I 



. , . sin e . n ,. . sin e . n ,. 



= cos e sin b -\ — sin (/ -\- b) -\- — — sin (/ — o), 



6 + & + S 



cos ■ 



a + 6 h 2 / _[_ 90° 



t an ___ - - e _ tan -31 



cos i — J 



The first equation is well known, the second is easily derived from 

 the known formula, expressed in the usual notation, 



A B sin (s — c) 

 tan — tan — = ^ -, 



2 2 sin s 



when we remember that, in considering the triangle, formed by the 

 heavenly body and the poles of the equator and ecliptic, A, B, s and 



c are replaced by 90° + a, 90° — I, 90° -f e ~ ^ + 8) and e. 



Suppose we were to tabulate the functions cos e sin A and ~— — sin A 



for a certain value of e (as 23° 27' 20" which is nearly its value at 

 present), and in small side tables put the variations of these functions 



for increments of 1", 2" 9" in e ; we should have the value of sin 8 



by entering the first table with the argument A = b, and the second 

 successively with the arguments A = I -j- b and A = I — b, and 



