REDUCTION FORMULA AND DETERMINATION OF CONSTANTS. 



Astronomical Work. 



The latitude, as already described in the previous section, is usually determined 

 from circummeridian altitudes of the Sun. In the field, the observer uses for his 

 preliminary computations of azimuth and time the latitude as derived from the 

 maximum observed altitude. The circummeridian reductions, as shown in the 

 specimen computation on page 31, are made at the office according to the formula: 



ha = h + Am -\- Bn + ... 



where ha = meridian altitude, h = observed altitude at the hour angle /, 



A = cos 5 cos ^ CSC To B = A"^ cot U 



2 sin^ \t 2 sin^ U 



in = . Ti n = 



sm 1 sm 1 



8 = meridian declination of the Sun or star 4> = latitude 



fo = meridian zenith distance of Sun or star.^ 



These formulae are not exact and must therefore be used within limits; in general, 

 the greatest value of t in minutes of time should not exceed one-half of zenith 

 distance in degrees. For an accuracy of 0'.05, and for values of the meridian zenith 

 distance not less than 10, the correction terms other than the first are negligible 

 within the limits given. For the computation of A, the approximate value of 

 latitude is usually sufficient. 



Outside of the stated limits, when the zenith distance is less than 10, as is fre- 

 quently the case in the tropics, the following exact formula is used in the ofhce 



reductions : 



. , ,, ,< cos </) cos 8 sin^ ht 



sm i(u - h) = ^Yi TT^ 



cos^{ho + h) 



Here / is as determined from the first reductions of time and azimuth." This re- 

 duction involves the assumption of the latitude for the first approximation to the 

 true value of the latitude; a second approximation is then made, using the value 

 so determined, and the operation is repeated until the assumed and computed values 

 agree. The formula does not give a very rapid approximation, but experienced 

 computers rarely have to make more than a second or third approximation. 



Azimuth and Time Computations. See specimen, pp. 32, 2>2). The zero of the 

 circle graduation having been referred to some terrestrial mark by sights before and 

 after the Sun work, the true azimuth of mark is determined from the computed 

 azimuths of the Sun for each mean time of observation by the formula: 



cot^ ^A = sec 5 sec {s p) sin {s //) sin {s </>) 



where A = Sun's azimuth reckoned from the south to the east for a. m. obser- 

 vations and to the west for p. m. work; h = mean observed altitude corrected for 

 refraction and parallax, = latitude, p = Sun's polar distance at the time of 



' Chauvenet, W. A manual of practical and spherical astronomy. Philadelphia. 1885, v. I. (235-250.) 

 * Idem, pp. 234 and 235. 



22 



