OF ECCENTRICITY AND GRADUATION. 27 



But each point of the curve from which the mean local corrections 

 were obtained depends upon two or more of the computed corrections: 

 the probable error of the correction applied to any angular measure- 

 ment is consequently less than 1.41 ~^ in some proportion depending 



upon the skill with which the curve was traced. This error is still 

 further diminished when the two readings differ so little that their mean 

 local corrections depend in part upon the same comparisons, and it van- 

 ishes when that difference is very small. For the series of total correc- 



t -t- 1".79 



tions in Table X, 1.41 -y— = 1.41 ~ .^ = -t- 1".45. 



VN 1/3 ~ 



The numerical mean of the residuals in Table IX is rb 2".7, which is so 



much greater than ~7=^ = ± 1".03, the probable variatiion due to error 



of observation, as to excite a suspicion that the sporadic errors of gradua- 

 tion, including systematic errors of short period, are rather large in this 

 instrument. 



The vernier is liable to errors as well as the limb, and they are some- 

 times large enough to require correction. If the initial and terminal 

 lines of the vernier do not simultaneously coincide with lines of the limb 

 the former is usually blamed, though not always justly^ for the vernier of 

 Sextant V in Table IV, if of the right length, will afford nearly simul- 

 taneous coincidences at the iuitial line, and at the additional line next 

 the terminal one, on all parts of the arc; while the vernier of Sextant VI 

 in the same table, should apparently be correct near 50°, too long at 0°, 

 and considerably too short at 140°. When the corrections of the limb 

 are known, a better judgment can be formed, but any examination by 

 comparisons of this sort is necessarily limited to the extremities of the 

 vernier, and gives no indication of the state of affairs between them. An 

 examination may be made with the sextant apparatus, however, at any 

 number of points, equidistant or otherwise, by bringing them in succes- 

 sion to the same division of the limb and comparing with the circle. 



Let I' be the actual length of a vernier, whose proper or intended 

 length is ^,* and whose nominal length, equal to one division of the limb, 

 is i; also let s be the reading at which the vernier is set in making a 

 comparison, r being the corresponding reading of the circle, and z an 



*The lengths I' and I are to be accounted positive when the readings of the limb 

 and vernier increase in the same direction — negative when they increase in oppo- 

 site directions. 



