440 



KRETZ. 



Also 

 SO that 



OS = tan X, 



X = tan X sin ?/ 

 Y = tan X cos 7] 



But from our spherical triangle 



sin X sin ?; = sin (a — a^) sin p. 



Dividing (5) by (i), and remembering (4) we get 



,. sin (a — fin) sin zJ cos ^ 



tan X sin ?/ = X=z ^ — 5*.' 1 £ _ 



cos (/g — q) cos p 



Similarly multiplying (2) by (6) we get 



,, cos (a — aA tan p tan { pn — a) 



tan X cos rj =: r=: !^ ^ ^-^^ — 



tan q 



These expressions may be easily transformed by the aid of 

 (3). We obtain finally 



(4) 



(5) 



(6) 



(7) 



X'- 



tan (a — a,,) sin^' 



where 



cos (A — ?) 

 F=tan(A-^) 

 tan q = tan p cos (a — a^) 



X8)i 



The formulae (8) express rigorously the relation which holds 

 between the true and the projected distances. They presuppose 

 a knowledge of the scale-value, and of the position of the center, 

 when the position of any other star may be found. 



From these formulae very convenient expressions can be ob- 

 tained in the form of series, giving the transformation correc- 

 tions to any desired degree of accuracy. They may be used with 

 advantage to within i 5° of the pole. Making the same assump- 

 tions as before with regard to the formulae (8), let us write them : 



X-- 



tan ( a 



cot q 



cos /q cot q -\- sin pQ 

 _ F tan A + I 



(9) 



(10) 



tan/o— ]' 



1 These are Turner's formula; for transforming measured rectangular into celestial 

 coordinates ; cf. Observatory, Vol. 16, pp. 373 ff. 



(100) 



