45 



observation is corrected by the appropriate formula to give 

 the azimuth of the star at elongation, so that practically we 

 obtain a series of observations at elongation instead of only 

 one. 



Notation. 



The following abbreviations will be used throughout : — 

 z denotes the zenith distance of the star in any position. 

 7? ,, ,, polar distance of the star. 

 A ,, ,, horizontal angle between star and pole. 

 I ,, ,, latitude of place of observation. 



c ,, ,, co-latitude of place of observation. 



h ,, ,, hour angle of the star in angular measure, 



t ,, ,, value of hour angle expressed in sidereal time. 



2q, Aq, h^, and f^, denote the values of z, A, h, and f 

 respectively when the star is at elongation. 



First Method, Horizontal Angle and Time being Noted 

 AT each Observation. 



In the spherical triangle having the star, the celestial 

 pole, and the zenith as its angular points, we have the follow- 

 ing fundamental relations : — 



cos A sin 2 = cos p sin c — cos c sin /; cos h ... (1) 



sin A sin 2 = sin p sin h ... ... ... ... (2) 



and from the corresponding right-angled triangle when the 

 star is at elongation 



sin p cos p cos h ^, 



sin 4o = = (3) 



sin c cos c 



cos A^^ = cos p sin h,^ ... ... ... ... (4) 



(l)x(3)-(2)x(4) gives 



sin z sin (.4^, — x4) = cos p sin p 2 sin^ J (h^^ — h) (5) 



This is an exact equation, but is unsuitable as it stands 

 for use in reduction cf observations, 

 sin p sin A 



. Putting = , (5) may be written 



sin z sin h 



sin (.4 - .4 ) 2 sin^ J (h^ - h) 



= cos J) 



sin .4 sin h 



2 sin' l(h,-h) 

 or, writing y = cos p — 



sin h 

 sin A Q cot .4 — cos Aq = 7/ 



Af^ is constant, and therefore .4 may be regarded as a 

 function of ?/. 



