12 THE CORRECTION OF SEXTANTS FOR ERRORS 



therefore, greater than that of a similar change affecting them both in the 

 same direction. Substituting in (2) the values of A and B from (7) : 



r — / = (- 1.5671 sin I •/ + 1.8384 (1 - cos i /)) [ D~\ (10) 



+ ^7.0467 sin i / — 11.0275 (1 - cos -i /)) [d sin ] S~\ 

 + (- 11.0275 sin i r' + 17.9311 (1 - cos \ ■/)) f/Xl-cosi^)! . 



The change in the computed correction for eccentricity resulting from 

 any given variation in [Z) sin 2 *S'] will be greatest when the coefficient 

 of the latter in (10) is a maximum^that is, when : 



7.6467 cos. \ r' — 11.0275 sin * / = 0, 

 tan \ r' = x^^' '-^"d / = 69° 29', 



the coefficient itself being then -|- 2.39. By similar means it is found 

 that the coefficient of [D (1 — cos \ Sy\ attains the numerical maximum 

 — 3.12 when / = 63° 1 V. Both coefficients reduce to for / = 0. If 

 the twenty-six products are correct to the second decimal place, the limit 

 of this error in the eccentric correction is, therefore, when / = 0, and 

 greatest when / is somewhere between 63° and 69°, but everywhere less 

 than 0."005 X 13 X (2.39 + 3.12) = 0."36. It will be shown a little 

 farther on that the probable error, due to errors of observation, in the 

 eccentric correction derived from a single series of comparisons, is when 

 y = 0, 0.42 t when y = 20°, and still greater for all larger values of 

 /, t being the probable error of a single comparison. As t can seldom 

 be much less than 2," this probable error is greater than the maximum 

 error in question. It is also apparent that to increase the possible error 

 tenfold, by retaining only one decimal digit in the products, would be 

 unsafe. 



The constants A, B, and A" are computed in the last column of Table 

 I. Each formula of (7) contains three terms Avhich may be obtained 

 from Table III without any greater inconvenience than that of taking 

 out the tabular products for each digit of the argument separately and 

 adding them together, but the table should be extended, if frequently 

 used. The headings of the columns refer to that argument which is in 

 the same horizontal line, e. g., the third column contains both the term 

 of B having [i> sin J S'\ for its argument, and the term of .4 for which 

 [Z> (1 — cos 2 S)] is the argument. The upper sign is to be applied 

 when the argument is positive, the lower if negative. These terms 



