IxXViii ASTRONOMY. 



and if the declination of the star is 8 and the latitude of the place <^, 



(l> = 8 ±z 



according as the star is south or north of the zenith. 



A more accurate application of the same principle is to observe the altitudes 

 of a circumpolar star at upper and lower culmination (above and below the pole). 

 The mean of these altitudes, corrected for refraction, is the latitude of the place. 

 This process, it will be observed, does not require a knowledge of the star's 

 declination, 



b. By the measured altitude of a star at a known time. 



/i =z measured altitude corrected for refraction, 

 Ts = observed sidereal time, 

 a, 8 = right ascension and declination of star, 

 / = hour angle of star, 

 </) = latitude of place. 



Then ^ may be computed by means of the following formulas : — 



t= Ts — a, 



o tan S sin ^ sin B 



tan B = ■ cos y = -. — j^-^, 



cos / sin 



In the application of these j8 may be taken numerically less than 90°, and since 

 / may also be taken less than 90°, /S may be taken with the same sign as 8. y is 

 indeterminate as to sign analytically, but whether it should be taken as positive 

 or negative can be decided in general by an approximate knowledge of the lati- 

 tude, which is always had except in localities near the equator. 



The most advantageous position of a star in determining latitude by this 

 method is in the meridian, and the least advantageous in the prime vertical. 

 When a series of observations on the same star is made, they should be equally 

 distributed about the meridian ; and when more than one star is observed it is 

 advantageous to observe equal numbers of them on the nortla and south of the 

 zenith. 



The application of this method to the pole star is especially well adapted to 

 the means available to the geographer and engineer, namely, a good theodolite 

 and a good timepiece. In this case the following simple formula for the latitude 

 may be used : — 



^■=^ h — p cos t -\- i/^ sin i" sin'^ / tan //, 



where/ is the polar distance of Polaris in seconds (about 5400"), and the other 

 symbols have the same meaning as defined above. Tables giving the logarithms 

 of/ and \p'^ sin i" are published in the American Ephemeris. 



