47 



A further considerable simplification would be made in 

 the reduction if it were possible to treat the denominator as 

 constant and write sin A^, instead of sin h. With any single 

 ■observation the error made, if this is done, may be considerable. 

 For instance, at a place in latitude 30°, if 'p = ?>0°, for an 

 observation made 15 minutes before elongation, the difference 

 made in the value of the second term, when sin /?„ is written in 

 the denominator instead of sin h, is about 5", whilst for an 

 observation made 30 minutes before elongation the difference 

 is about 35". But, if we have a series of fairly well-balanced 

 observations made both before and after elongation, the 

 values of h range fairly evenly on each side of ll^^, and on 

 averaging up the set there will be very little difference 

 whether we use h or h^^, the difference being generally of the 

 order of 1". So that in such a case it is usually quite sufii- 

 ■cient for the surveyor to use h^ instead of h. We may then 

 make a further slight simplification by putting 



sin A Q cos p 



= tan A ^ cos^ p 



sin Jiq 



Practical Computation. 



We therefore conclude that, for the ordinary work of the 

 surveyor, a series of well-balanced observations extending to 

 about half an hour on each side of elongation on any circum- 

 polar star may be reduced to a series of observations at 

 elongation by the formula 



2 sin2 1 {h^ - h) 



A - ^4 = tan .4 (, cos^ ;; "- ... (6) 



sin 1" 



in which .4^, — .4 is given in seconds of arc. 



If, however, only one or two observations are to be 

 reduced, as may be the case if the star at elongation has been 

 obscured by clouds, or the observations are badly balanced 

 and have been made mostly on one side of elongation, or if 

 the greatest possible degree of accuracy is required in the 

 computations, the formula used should be 



cos jJ 2 sin- J {h^ — h) 



.4,-.4=sin.4, — ... (7) 



sin h sin 1" 



Tliis form may be obtained directly from (5) by considering 

 Af^—A as a small angle so that the sine may be written equal 

 to its circular measure. 



