54 



It follows, therefore, that for the ordinary work of the 

 surveyor the correction involved in the last term of the series 

 is quite negligible for observations extending over a range of 

 altitude of 2°, or 1° on each side of elongation, provided that 

 the star does not pass within 10° of the zenith. At places 

 near the equator the observations may clearly extend over a 

 very much greater range of altitude with the same degree of 

 precision. 



To determine over what range of time the observations 

 may extend, we find on differentiating the equation 



cos z = cos c cos jj -f sin c sin p cos h 



d z sin c. sin p sin li 



that = ■= sin j) for a star at elonga- 



d h sin z 



tion. This = i, if :p = 30°. 



Thus the rate of change of altitude at elongation does not 

 depend on the latitude, but simply on the polar distance of 

 the star, and for a star distant 30° from the pole we have 



d h =1d z 



Therefore, if d z — 1°, d h = \20' of arc, or 8 minutes of 

 time, the altitude of the star near elongation thus changes 

 by 1° in about 8 minutes. For stars closer to the Pole the 

 time taken for the same change of altitude will be greater. 



Practical Computation. 

 We conclude that for a set of observations extending over 

 a range of altitude of about 2°, or 1° on each side of elonga- 

 tion, occupying, in the case of a star with a polar distance of 

 30°, about 16 minutes of time, it is amply sufficient to use the 

 formula 



cot ;; {-^-^y 



A^^-A= sin 1" (10) 



sin 2„ 2 



It should be noticed that the error made by the use of 

 this formula in the final reduction of a set of observations will 

 be very much less than the error made in the reduction of 

 the single observation furthest from elongation. We have 

 based the stated limitations upon the error made in the reduc- 

 tion of the single observation, so that for a complete set of 

 observations the time occupied may be extended somewhat 

 beyond the limits given above. In low latitudes the observa- 

 tions may extend over a greater range than in high latitudes. 

 In latitude 10°, for instance, the observations may extend 

 over half an hour, and formula (10) will still give the average 

 result of the set of readings correct within less than 1". 



