26 



THE CORRECTION OP SEXTANTS FOR ERRORS 



Table X. 



85°, and again too small from 85° onward. From a point near 68°, 

 where it is greatest, their length decreases in both directions. These 

 systematic irregularities may have been produced by inequalities in the 

 operation of the dividing engine, but they can also be accounted for by 

 supposing an almost infinitesimal distortion to have occurred after the 

 graduation was executed, the middle of the arc approaching the center, 

 and the ends receding therefrom, the greatest change being at the end 

 opposite 0°. The apparent difference between the 

 two ends may, however, be partially due to a slight 

 deviation from parallelism in the surfaces of the 

 index glass. 



The total correction of this sextant, or sum of the 

 eccentric and mean local corrections, is given in 

 Table X. For the sake of convenience in use the 

 correction at 0° has been reduced to by adding 

 — 3" throughout. 



The correction applied to any angle measured with 

 the sextant is always the difference between two tab- 

 ular corrections — that of the observed reading, and 

 that of the reading made in determining the index 

 correction. Let D/, 2)/, etc., be the observed values 

 of D corresponding to the setting S' in the different 

 series of comparisons, the computed local correction 

 for this reading is then : 



D; — A sin i S' — B(l — cos i S') — X,-^] 



Bl _ ^ sin ^ S' 

 etc., + etc., -f 

 X>;_^sin^>S' — ^(1 



2- X>' — 2' X 



J5 (1 — cos h S') — X.,-\- 



} ^N = 



cos A .5') --^ J 



iV 



— A sin h^' — B {1 — cos i S'), 



and the sum of the eccentric and local corrections is : 



N 



For 



2" J)" IX 



any other setting /S" this sum is: ^^^ — , which subtracted from 



iV 

 the jireceding expression leaves *. 



Z 1/ — I D". 



Now the probable error of each of the N differences D{, D/', Z),/, Z),", 



I D' — I B" 



etc., is t; the probable error of 



N 



is therefore : 



N VN 



