Ixviii ASTRONOMY. 



These quantities are usually designated by the following notation : — 



A = the azimuth of a star or object, 

 A = its altitude, 



z = its zenith distance = 90° — /', 

 / = its hour angle, 

 8 =: its declination, 

 j> = its polar distance = 90° — 8, 

 ^ = the parallactic angle, or angle at the star between the pole and the 



zenith, 

 <f> = the latitude of the place of observation. 



b. Altitude and azimuth in terms of declination and hour angle. 

 The fundamental relations for this problem are — 



sin // = sin <^ sin 8 -f- cos ^ cos 8 cos /, 

 cos // cos y^ =: — cos <^ sin 8 -}- sin ^ cos 8 cos f, (i) 



cos /i sin A ^ cos 8 sin A 



When it is desired to compute both A and /i by means of logarithms, the most 

 convenient formulas are, 



m sin M = sin 8, ^ i^ t^" ^ 



? * tanJ/=^-— .» 



m cos M = cos 8 cos t, cos t 



tan / cos M , ^ 



sin /i = m cos (<^ — J/). tan A = gin {^ _ J/)' ^^ 



cos // cos A = m sin (</. - Jif), ^^^ j^ ^o^ ^ . 



cos // sin ^ = cos 8 sin /, t^n (<^ - M) 



A > i8o° when / > i8o° and A < i8o° when t < i8o°. 



For the computation of A and z separately, the following formulas are useful : 



sin / 



tzn A = — cos ^ tan 8 (i — tan ^ cot 8 cos /) 



(3) 

 <^ sin / 



~ 1 — l> cos /' 



where 



a = sec c^ cot 8, <^ = tan ^ cot 8. 



Formulas (3) are especially appropriate for the computation of a series of 

 azimuths of close circumpolar stars, since a and d will be constant for a given 



place and date. 



cos z = cos (^ ~ 8) — 2 cos (f) cos 8 sin'^ h f, 



sin^ i 2 = sin'^ i (<^ ~ S) + cos </> cos 8 sin" ^ /, ^^-^ 



((/> ~ 8) = <^ — 8, f or </) > 8 



= 8 — <^, for </)< 8. 



