46 



Differentiating, we have 



1 d A 



— sin A^ = 1 



sin^ .4 d y 



^2 A . d A 

 and — sin .4,, = 2 sin .4 cos A 



d. y~ d y 



Therefore, when y — o 



d A d^ A 



= — sin A and = sin 2 .4 „ 



dy dy^ 



and consequently, by Taylor's Theorem 



cos jp 2 sin^ \ (7? ,5 — h) cos^ ^ 2 sin'* \ {h^ — h) 

 A =AQ — sin A (, ■. 1- sin 2 A ^, 



sin h sin 1" sin- h sin 1" 



provided that A^ — A is measured in seconds of arc. 



Tills is a convenient converging series for the determination 

 of the difference between A and .4^,, in which the terms 

 diminish so rapidy that in all ordinary work it is not neces- 

 sary to take into account any term except the first. Thus, 

 if the observations are made at a place in latitude 30°, on 

 a star with a polar distance of 30°, and are continued for 

 fifteen minutes of time on each side of elongation, the extreme 

 value of A — 7iq = 3° 45'. The corresponding value of the first 

 term in the series then works out at 229", or 3' 49", and that 

 of the second term at less than J". If ^ — i'„ = 30 minutes, 

 or h — h^^ — 7° 30', then under the same conditions the first 

 term = 902" and the second term only 5|". With the same 

 polar distance and in the same latitude, the limiting value 

 for t — f^, in order that the second term may not be greater 

 than 1", is about 19 minutes. On repeating the calculations 

 for a place in latitude 20°, and again for a place in lati- 

 tude 40°, it is found that in neither case does the limiting 

 value of t — t^ differ by more than a minute from the value 

 previously found if the second term in the series is to be 

 less than 1". 



It thus appears that, even if the mathematical reduction 

 of each single observation is to be correct within 1" of arc, 

 it is sufficient to use only the first term of the series if the 

 observations extend over a period of about 19 minutes on 

 each side of the elongation. The average of the whole series 

 may be correct within this limit, even if the time extends 

 over a considerably longer period, because the error in reduc- 

 tion will exceed 1" only in the case of the extreme observations. 



