228 CATALOGUE OF POLAR STAKS. 



Berliner Jahrbuch. For the years 1871 to 1879 inclusive, they were obtained by 

 applying to the yearly epheiuerides of the Gesellschaft the corresponding correc- 

 tions by which the provisional system is reduced to the system of Publication XIV. 



For the places of the fundamental stars between 1860 and 1870, and fur the 

 places of all non-fundamental stars for the entire period between 1800 and 1885, 

 the reduction-elements given in the Harvai'd College Catalogue were employed. 



For stars below 85" north declination the development of a and S in terms of 

 the first, second, and third powers of the time will be sufficiently accurate for the 

 limit of fifteen years. For the reduction of stars near the pole, the problem becomes 

 more difficult. Since the method of development by differential coefficients in 

 terms of the ascending powers of the time has necessary limitations in its applica- 

 tion, it has been thought advisable to give an illustration of the various methods 

 by which the co-ordinates for any time ^ are reduced to those for any time f . The 

 star Groombridge 1119 is selected for this purpose. Tlie reductions for precession 

 and for proper motion will be considered independently. 



Reduction of the RUjht Ascension and the Declination fur the JEjuator and Equinox of any Time 

 t^ to the Values for any Time t' by the Trigonometrical Method of Bohnenheryer. 



From Bessel's TabultB Regiomontanaa, pp. vii, viii, we have 



cos 5' sin («' + ;i' - z') = cos 8 sin {a + I + z) 



cos <5' cos (k' + /.' — z') = cos 8 cos (« + I + z) cos — sin 8 sin U (1) 



sin 8' = cos 5 cos (a + 1 + s) sin 6 + sin 8 cos 

 Writing for brevity 



A = u + I +z. A' ^ «' + ;.' - z', 



we have 



cos d' sin A' = cos 5 sin ^ 1 



cos 8' cos A' = cos 8 cos A cos d — sin 8 sin d \ (-2) 



sin 8' = cos 8 cos A sin + sin 8 cos J 



Fi-om the first two equations of (2) we obtain 



cos 8' sin (A' — A) = cos 8 sin A sin [tan 8 + tan h cos A'] 



cos 8' cos {A' — A) = cos 8 — cos 8 cos A sin [tan 8 + tan ^ cos A^ . 



If we put 



p =^sin [tan 8 + tan \ cos ^1], 

 we have 



(3) 



and 



tan (A' - A) = zr^ j' (V 



^ ' \ — p cos A 



cos \ {A' + A') /c\ 



tan i (5' - c-*) = tan I ,,, ^ \^, _ ^\ - ('^ 



