OF ecx;entricity and graduation. 9 



as to rendet- this observatiou of doubtful utility. A single series of such 

 comparisons furnishes the eccentric correction with sufficient precision for 

 a sextant of the second class, while the residuals, containing the errors 

 of both graduation and observation, afford a trustworthy indication of the 

 performance to be expected of the instrument under very favorable cir- 

 cumstances. It was proposed to subject sextants of the first class to an 

 examination comprising several similar series of comparisons, made with 

 diflTerent portions of the circle, the number depending somewhat upon the 

 circumstances of each case. These repetitions yield not oidy improved 

 values of the corrections for eccentricity, but also a means of separating 

 the local errors of graduation from the errors of observation. 



The normal equations obtained by making M= 14, and -iS successively 

 0°, 10°, 20°, etc., 130°, in (5) are: 



4.72172 A + 2.1794G B + 7.06526 X — ID sin h S'] = 0, 1 



2.17946 A + 1.09780 B + 2.90976 X — \_D (1— cos J Sy] = 0, V (6) 

 7.06526 A +.2.90976 B+ 14 X - [D] = 0, J 



from which are deduced : 



^1 = -1.5671 [/>]+ 7.6467 [Z) sin ^S]- 11.0275 [Z>(1- cos A /S)],] 

 B= 1.8384[i)]-11.0275[Z)sin J /SH 17.9311 [Z)(l- cos i>6')], \ (1) 

 A= 0.4802 [D]- 1.5671 [d sin I. S^ + 1.8384 [Z>(1 - cos h S)], J 



and also : 



A' = 0.0714 [D] - 0.5047 A — 0.2078 B. (8) 



From each observed value of D let the corresponding value computed 

 by (4) be subtracted ; the remainder, which may be designated — C, 

 is the sum of an observed local correction of the graduation, and an error 

 of observation. Any single observation made under circumstances as 

 favorable as those attending the examination, and corrected for eccen- 

 tricity only, will, therefore, contain an error the most j»robable estimate 

 of which is : 



0.6745 ^/tC^^=-^ = i^/o.04136[(0- C)^], 



(9) 



if the supposition is made that the errors of graduation are either small 

 enough to be neglected, or else, like the errors of observation, devoid of 

 any systematic arrangement. This assumption cannot always be abso. 

 lutely correct, but no other is eligible when only one series of comparisons 

 has been made. 



The following example (Table 1) exhibits a convenient form of record 

 and computation, requiring no tables except Crelle's Recheutafeln, and 

 those contained in these pages. The entire calculation is given here. At 



