THE ORBITS OF T II E EIGHT PRINCIPAL PLANETS. 185 



The equations for reducing from longitude and latitude to right ascension and 

 declination, and the reverse, are the following: — 



cos b cos a= cos t 3 cos A \ 



cos b sin a= cos e cos (3 sin "k — sin e sin (} f (584) 



sin S= sin e cos (3 sin X-(-cos e sin /3 J 



cos /3 cos A=cos b cos a } 



cos (3 sin A=eos 6 cos b sin a-(-sin e sin b \- (5S,"}) 



sin |S= — sin e cos 5 sin a-|-cos.e sin b J 



Example IV. The right ascension and declination of a Tauri {Aldeharan) in 

 1850 was a=66° 49' 46".35, ^= + 16° 12' 11". 0; required its longitude and latitude 

 for *=— 4900, or, for the beginning of the year B. C. 3050. 



Since e=^2S° 27' 31".0, in 1850, equations (585) will give the longitude and 

 latitude for the same epoch, as follows : — 



A=67° 41" 34'.1, (3=— 5° 28' 40". 1. 



And for /=— 4900. Tables VIII and IX will give 



G"=5° 22' 51".7, <?>"=0 o 41' 22".45, and $=— 67° 40 32".2. 



If these quantities be substituted in equations (573), we shall find 



X— 6"— <//=62° 16'45".5, sin (3'=— 0.106061 ; 

 whence we get 



Jl'=359 59' 5".0, and /5'=— 6° 5' 17".9. 



Therefore the star Aldeharan, at the beginning of the year B. C. 3050, was only 

 55".0 westward of the vernal equinox, measured on the ecliptic of that date, and 

 coincided with the equinox in the year 3049 B. C. 



Example V. The right ascension and declination of Aldeharan being, as in tbe 

 preceding example, at the beginning of 1850, required its right ascension and 

 declination at the beginning of the year B. C. 3050. For 1= — 4900, Table X 

 gives z=— 31° 43' 26".4, ^=—32° 44' 31".8, 0=— 26° 14' 1".4, and 3'= + l° 31'0"(). 

 If these quantities be substituted in equations (580), we shall find 



a'— J— $'=33° 42' 2".8, and sin_5'=— 0.0969604. 

 Whence we get 



a'=2° 28' 31".0, and b'=— 5° 33" 51'.0. 



We might have found these last quantities by means of equations (584) by sub- 

 stituting for a,, (3, the values of '/! and |S', found in example IV, and using the 

 value off corresponding to that epoch, which value is g=24° 3' 8\2. 



From this computation we sec that, although the star Aldeharan was 55" westward 

 of the equinox when measured on the ecliptic, it was nearly 2g degrees eastward 

 of the equinox when measured on the equator, and instead of being in a northern 

 constellation then, as now, it was in reality in a southern constellation. 



11. We will now determine the position of the pole of the equator. The longi- 

 tude of the pole on the fixed ecliptic of 1850 at any time t will evidently be equal 



24 April, 1S72. 



