and Declination into Longitude and Latitude. 333 



signs. In these equations x', 8' and y for a constant s are 

 functions of a only. If, therefore, X', V, sin y, cos y, be re- 

 duced into tables with single entry, the argument being a, it 

 will only be required to calculate, 



tang j3 = tang (5 — 8') cos y 

 tang /3 = tang (S — 8') sin y cosp 

 X = X' + p. 

 The geometrical signification of these auxiliary quantities 

 will be found without trouble. 



Exactly in the same manner vve have for system (2), 

 tang «' = tang X sec s 

 tang (3' = sin X tang s 

 cos y' = cos X sin s 

 sin 8 = sin y' sin (/3 + /3') 

 cos 8 sin («' — «) = cos y' sin (/3f /3') 

 cos 8 cos («'—«) = cos ((3-1-/3') 

 If, therefore, the following quantities are reduced to tables 

 whose argument k : 



tang A = tang k sec s 

 tanff B = sin k tang s 

 a = cos k sm e 



b = 



cos B 



we shall find for k = u 



tangp = a tang (8— B) \ = A+p 

 tangf3 = b tang (8 — B) cos^ 



and for ^ = x 



tang q = a tang (/3 + B) a = A — y 

 tang 8 = 6 tang (/3 + B) cos g- 

 With respect to the purpose for which the tables are here in- 

 tended, viz. converting the geocentric « and 8 into X and /3, 

 for planets or such heavenly bodies as are within the zodiac, or 

 for which /3 < + 10°, it will be sufficient, if an approxima- 

 tion is only required, to put: /3 = Z» (8 — B) 



X = A + « (8-B) sec^ 

 where even the factor sec ^ may in most cases be neglected, 

 because its value is only, 



The 



