22 THE CORRECTION OF SEXTANTS FOR ERRORS 



The three terms of each group in the fifth colunui are obtained by suc- 

 cessively addiug X', X", and X'", to the eccentric correction in the pre- 

 ceding column. The three values of Z> following in the sixth column are 

 those which were observed in the first, second, and third, series of com- 

 parisons respectively. By subtracting from each of the latter the com- 

 puted value in the fifth column, the three values of the local correction in 

 the seventh column are found ; their mean is given in the eighth column. 



The residuals in the ninth column are obtained by subtracting this 

 mean successively from its three constituents, and the tenth column con- 

 tains the squares of the residuals. 



There are 42 residuals, and the sum of their squares is 260 ; the prob- 

 able error of a single observation is therefore : 



t = 0.6745 x/,^^-^ = ± r.79. 

 \ 42 — 5 



The probable errors of the eccentric corrections in the fourth column of 



Table VIII are taken from Table V with this value of i, and divided by 



l/3, since three independent determinations of A and B have been made. 



The maximum jirobable error of a local correction deduced from a 



single series of comparisons, as given in the second column of Table VI, 



is i X 0.93 ; the probable error of the local corrections in this example 



(excepting that derived from the additional comparison at 140°) is there- 



-1- 1" 79 X 93 

 fore not greater than '- — ~ — '- — = ± 1".0, which is small enough to 



t/3 

 justify some degree of confidence in them. It should be mentioned here 

 that the three series of comparisons were all made with the same portion 

 of the circle in this instance, and that the effect of errors in the circle is 

 consequently but little diminished by the repetition. In a mere illustra- 

 tion of the capabilities of the method this uniformity is preferable, since it 

 affords a value of t nearly identical with that which would be obtained 

 if the circle were faultless, while the absolute verity of the corrections is 

 of minor importance. But charging the sextant with the imperfections 

 of both instruments, and ignoring also the error of observation, which 

 cannot be inappreciable, none of these corrections imply an error of cir- 

 cular division exceeding 5", one that is certainly to be expected in all 

 graduation except that of the very highest class. 



The probable error of observation in this example, ^ := dz 1".8, is very 

 small, as it ought to be, for the sextant was firmly supported in a con- 

 venient position, the pointing was deliberate, and directed upon a singu" 

 larly well-defined object, the index was set in a definite position always 

 referred to the same lines of the vernier, and the observer was perhaps 

 somewhat expert at that time. This error will ordinarily be larger, indeed 



