332 Prof. Encke on the Conversion of Right Ascension 



The formuUe (1) and (2) may be venderetl convenient for 

 logarithmic calculation in another way, by assuming for each 

 of the two systems a particular case as a foundation. 



Designating in (1) by x' and V the longitude and declination 

 of a point whose right ascension = «, and latitude = 0, we 

 have = cos e sin 8' — sin e cos S' sin « 



sin a' = sin e sin V + cos £ cos V sin a (3) 



cos a' = cos a cos S' 



whence sin x' sin s = sin V 



sin x' cos £ = cos I' sin a. (4') 



cos x' = cos 8' cos a. 



Combining (1) and (3) by multiplying with cos 8 and cos 8', as 

 also with cos x' and sin x', we have 



sin/3 = -^ sin (8 -8') 



' cos d ^ ' 



cos /3 sin (X — X') = sin e cos a sin (8 — 8') 

 cos |3 cos (X — X') = cos 8 cos 8' cos «-+ sin £^ sin 8 sin 8' 

 + cos £^ sin a? cos 8 cos 8' 

 + sin e cos £ sin « sin (8 -f 8') 

 where the latter formula may also be written thus : 



cos ^ cos (X-X') = cos (8-8')_ 



— { cos £ sin 8 — sin e cos 8 sin « } x 

 {cos £ sin 8' — sin £ cos 8' sin «} 

 and as by the first equation of (3) the latter part is always = 0, 

 we have . cos e . ^ j,. 



cos 



COS /3 sin (X — x') = sin e cos a sin (8 — 8') (5) 



cos /3 cos (x — x') = cos (8 — 8') 



These three equations squared, give 



1 = J i^ + sin £- cos «^ \ sin (8-8')^ + cos (8-8')^ 

 It will, therefore, be allowed to put 



cos t 



= sm y, sm £ cos « := cos y 



cos S' 



where on account of the first eciuation of (5) y must always 

 be assumed less than 180°. The calculation is thus reduced 

 to these six equations : 



tang x' = tang « sec e 

 tang 8' = sin « tanjj s 

 cos y =: cos « sin = 

 sin ^ = sin y sin (8 — 8') 

 cos /3 sin (X — x') = cos y sin (8 — 8') 

 cos /3 cos (X — x') = cos (8 — 8') 



in which cos x' and cos « must always be taken with equal 



siirns 



