Ixxvi ASTRONOMY. 



1. Observe time of transit of a star south of zenith, 



2. Observe time of transit of a star north of zenith. 



Reverse telescope, 



3. Observe time of transit of another star south of zenith, 



4. Observe time of transit of another star north of zenith. 



Each star observation will give an equation of the form (i), and the mean of 

 the four resulting equations is 



' 4 ' 4 ' 4 4 ■ 



Now the coefficient of r in this equation may be always made zero by taking 

 for the epoch 7^ the mean of the observed times T. Likewise, ^A and 2C may 

 be made small by suitably selected stars, since two of the A's and Cs are positive 

 and two negative. The value \ tia. — T) is thus always a close approximation to 

 AZfor the epoch Tq = \ %T, when %A and 2C approximate to zero. But if these 

 sums are not small, approximate values of a and c may be found from the four 

 equations of the form (i), neglecting the rate, and these substituted in the above 

 formula will give all needful precision. 



For refined work, as in determining differences of longitude, several groups of 

 stars are observed, half of them with the telescope in one position and half in the 

 reverse position, and the quantities AT", a, c, and r are computed by the method 

 of least squares. In such work it is always advantageous to select the stars with 

 a view to making the sums of the azimuth and coUimation coefficients approxi- 

 mate to zero, since this gives the highest precision and entails the simplest com- 

 putations.* 



b. By a single observed altitude of a star. 



An approximate determination of time, often sufficient for the purposes of the 

 geographer, may be had by observing the altitude or zenith distance of a known 

 star. The method requires also a knowledge of the latitude of the place. 

 Let 



Zi = the observed zenith distance of the star, 

 /? =. the refraction, 

 z = the true zenith distance of the star, 



= ^1 + ^^, 

 a, S, = the right ascension and declination of the star, 



/ = hour angle of star at time of observation, 



T=z observed time when Si is measured, 



AT^ correction to timepiece, 



(f) =z latitude of place. ■» 



Then the hour angle / may be comjDUted by 



„ , sin (a- — (h) cos (o- — S) 1 / 1 I 5^ I \ 



^ cos o- cos {a- — z) 



* For details of theory and practice in time work done according to this plan see Bulletin 49, 

 U. S. Geological Survey. 



