188 Mr. William Galbraith, on some 



All the formulae* with which I am acquainted, and most 

 of the tables are adapted to the sun's distance from the 

 solstice reckoned on the ecliptic, or the difference between 

 the sun's longitude, at the time of observation, and 90° or 

 270°. Now, by those possessing an ephemeris giving the 

 sun's longitude at apparent noon, with differences to reduce 

 to any given meridian, this is readily found. The sun's 

 longitude, however, in the new Nautical Almanac for 1834, 

 and succeeding years, is given to mean noon without dif- 

 ferences or proportional parts, consequently, the distance 

 of the sun, at apparent noon, from the solstice is not so 

 easily obtained in terms of the longitude, as in those of the 

 right ascension. Besides, in an observatory, the sidereal 

 time is generally known by observation, and, therefore, on 

 the whole, arguments depending on the right ascension are 

 the more convenient for obtaining the reduction of the 

 sun's observed declination to the solstice. 



A very convenient formula for this purpose, may be ob- 

 tained in terms of the right ascension as follows : 



Let K be the right ascension at the time of observation, 3 

 the declination, w the obliquity of the ecliptic, and x the 

 connexion necessary to reduce the observed declination to 

 the solstice. 



By spherics, sin. /c tan. lo = tan. ^ = tan. (w — x). 



_, ^ ^ tan. w — tan. x . „ 



But tan. (w — x) = -r-rz 1 " tnereiore, 



^ ^ 1 + tan . w tan . x ' 



tan. w — tan. x 



sm. K tan. w =i-— ~i : 



•^ l-|-tan.i(;tan.a7 



which by reduction becomes, 



(1— sin. k) tan. w 



tan. X = 1— — -. '- ^p- (8) 



A + sin. K tan. 2 w ^ ^ 



This equation would give the reduction to the solstice, 

 but it is not in a form to be readily applied. It admits of 

 a transformation, however, from the following considera- 

 tions, which renders it remarkably simple. Since k does 

 not in this case differ much from 6^ or 18^, let a = 6^ — k, 

 K — 6^, 18^ — fc, K — 18**, and a being small cos. k- = 1 — 

 A* A* A6 



~2~ "^24 — 720 "^ If this value of COS. K be sub- 



♦ There are, I have since found, formulaj, though still requiring simplification, 

 iu some works on Astronomy for this purpose, and not free from obliquity. 



