IXXX ASTRONOMY. 



a given star and date ; and hence they need be computed but once for a series of 

 observations on the same star on one date. 



The effects of small errors A/, A^, and AS in the assumed time, latitude, and 

 declination are expressed by 



cos 8 cos q ^ , . ^ . . sin <7 „ 



-. A/, — sm A cot z Ad), -. — ^ AS, 



siu z ' ^' sin z ' 



respectively, where z and q are the zenith distance and parallactic angle of the 



star. Hence the effect of A/ will vanish for a star at elongation ; the effect of 



A^ vanishes for a star in the meridian, and is always small (in middle latitudes) 



for a close circumpolar star ; the effect of AS vanishes for a star in the meridian. 



It appears advantageous, therefore, to observe for azimuth (in middle latitudes) 



close circumpolar stars at elongations, since the effect of the time error is then 



least, and the eft'ects of errors in the latitude and declination are small and may 



be eliminated entirely by observing the same star at both elongations. 



The hour angle 4, the azimuth A^, and the altitude //^ of a star at elongation 



are given by the formulas (2) of section 2,/! Those best suited to the purpose 



are 



K"" — sin (8 + <^) sin (8 - <^), 



K ^ cos 8 , sin </> (2\ 



tan 4 = "= — 1 ^' tan A^ = r. > tan h. = — pr— v / 



* sm <^ cos 8 ^ A ^ K 



Knowing the time of elongation of a close circumpolar star, it suffices for many 

 purposes to observe the angle between the star and some reference terrestrial 

 mark at or in the vicinity of that time. 



For precise determinations of azimuth it is customary to observe a star near 

 its elongation repeatedly, thus obtaining a series of results whose mean will be 

 sensibly free from errors of observation and errors due to instrumental defects. 



The computation of the azimuth A may be made accurately in all cases by the 

 formulas (i) ; but when a close circumpolar star is observed near elongation, it 

 may be more convenient to use the following formulas : — 



M =: (J — 4), or the interval before or after elongation at the time of 

 observation, 

 ^A = (A — A^), or the difference in azimuths of the star at the time 



of elongation and at the time of observation, (3) 



^^ — 2 p" sin 4 cos </> ^^^^ =^ 2 (p"y sin 4 tan 4 cos </, ^^^ ^ 



8# 



* To the same order of approximation one may write in the first term of this expression 



(1=;)- , ,. . o - sin2^ A^ 



^ {Aisy2 = p" 2 sin2 i Ai = -^--^-' 



(2 sin" i Ai) 

 which latter is the most convenient form when tables giving log - /> — - for the argument A/ 



in time are at hand. Such tables are given in Chauvenet's Manual of Spherical and Practical 

 Astronomy (for full title see p. Ixxxii), and in I'ormeln und HiUfstafeln fiir Geographische Oris- 

 bcstimmungc7i, von Dr. Th. Albrccht. Leipzig: Wilhelm Engelinann, 4to, 2d ed., 1879. 



