53 



Differentiating the equation three times in succession, the 

 work being rather long but quite straightforward, we find 

 that when x = o 



d A 



— o 



d or 





d-" A 



cot p 



d x^ 



sin 2;, 



d' A 



3 cot p cos z^^ 



d x"" 



sin- 



Tlierefore by Taylor's Theorem 



CQ)tp{z — Z^^ COt^COS^o (2 — Z(i)' 



A=A,- 



sinl" + 



sin^ 1 



(9) 



sin Zq 2 sin^ z^ 2 



provided that A^ — A and z — z^^ are expressed in seconds of arc. 



To get some idea of the relative values of the terms in 

 this series, we find, if the star observed has a polar distance 

 of 30° and the latitude is also 30°, then 2^ = 54° 44' 09", and 

 if 2 — 2q = 1°, the second term works out at 66" and the last 

 term to 0'8". If z^0, = 2° the values become 264" and 6" 

 respectively. 



The last term in (9) is equal to 



cos"' p cos c 



X sin^ 1" 



sin p (cos^ p — cos^ c) 2 



and has therefore an infinite value if p = c, in which case the 

 star passes through the zenith. This is clearly of no practical 

 importance. 



The following are the values of the last terms in different 

 latitudes for a star 30° distant from the celestial pole, if 



z- 



-z, = r: — 





Latitude. Value of last term in (9). 





50° 3-5" 





40° 1-5" 





30° 0*8" 





20° 0*4" 





10° 0-2" 





0° 0" 





If 2 — ^o = 2° the preceding values should be multiplied 



by 8. 



