ASTRONOMY. 



Ixxxi 



This last formula gives Av4 in seconds of arc when A/ is expressed in seconds 

 of time ; A/ is considered positive in all cases (in the use of the formula), and 

 with this convention the positive sign is used when the star is between either 

 elongation and upper culmination, and the negative sign when the star is between 

 either elongation and lower culmination. For a given star, place, and date the 

 coefficients'^of (M'f and (^ff will be sensibly constant and their logarithms will 

 thus be constant for a series of observations of a star on any date. By reason of 

 the large factors (p" = 206 264."8)' and tan 4 in the denominator of the second 

 term, it" will be very small unless A/* is large. Hence this term may often be 

 neglected. Using both terms, the formula will give A^ for Polaris to the nearest 

 o."oi when A/" < 40'" and when observations are made in middle latitudes. 



By reference to formulas (2) of section 2,/, it is seen that 



sin ?> cos 8 sin^ 8 cos 8 



sin 4 cos </> J^ 



sin 8 cos 8 sin'^ 8 cos^ 8 sin <j> 



sin /e tan 4 cos ^ ^^ 



X^ = sin (8 + <^) sin (8 - <^).* 



b. By an observed altitude of a star. 



/% = true altitude of star observed ; /. e., the observed altitude less the refrac- 

 tion, 

 cf, = latitude of place, 

 p = polar distance of star, 

 A := azimuth of star. 



tan^ i,A = sin (o--.^) sin (^-/Q 



cos cr cos ((T — /) 



The most advantageous position of the star, on account of possible error in the 

 observed value of /i, is that in which sin A is a maximum. I1iis position is then 

 at elongation for stars which elongate, in the prime vertical for stars which cross 

 this ereat circle, and in the horizon for a star which neither elongates nor crosses 

 the prime vertical. A star will elongate when /> < 90° — c/) ; it will cross the 

 prime vertical when/ lies between 90° — <^ and 90° ; and it will neither elongate 

 nor cross the prime vertical when/ >9o°, or when the declination (8) of the star 

 is negative. 



c. By equal altitudes of a star. 



By this method, when a fixed star is observed first east of the meridian and 

 then west of the meridian at the same altitude, the direction of the meridian will 



* In precise work the computed azimuth requires the following correction for daily aberration, 



namely : — 



cos <p 

 AA = — o. '52 —. — cos A, 

 "^ sm z ' 



where A is to be reckoned from the south by way of the west through 360°. 



