30 THE CORRECTION OF SEXTANTS FOR ERRORS 



examinatiou and computation. If comparisons are made at the two ends 

 of the vernier, and at nine equidistant points between them, -. is succes- 

 sively 0, 0.1, 0.2, etc., 1.0 ; m= 11, and the normal equations (11) become : 



3.85 c + 5.5 X— r-dl =0,] 



^'^ [ (13) 



I 

 J 



5.5 c + 11 a; — frfl =. 0. 



Their solution gives : 



c = — 0.455 



ffzi + 0.909 r^n,! 



1- (14) 



0.318 fdl —0.455 r';(/1, 



and the weight of c is 1.1. 



Table XI is an example of the record of examination and form of com- 

 putation. It refers to a sextant divided to 10', and reading to 10" by a 

 "short" vernier, for which, therefore, i = 10' and ^9 =:= 59. The calcula- 

 tion scarcely requires any tables except one of squares for extracting the 

 square root in finding the probable errors, but it is convenient to take 

 from Crelle's table the four products employed in computing the values 

 of c and x by (14), 



The first three columns contain the record of examination, and the fourth 

 receives the products ps, when they are expressed. For the sake of clear- 

 ness the values of r have been given in full, as well as the seconds of ps, 

 but it is unnecessary to write down more than the seconds of r, and the 

 seconds of ps when they vary, as in the case of a sextant divided to 15' 

 and reading to 15". A sufficient explanation of the remaining columns 

 is to be found in their headings. The sum of the squares of the residuals 



8fi 



n o"^ 



± 2". 09 

 -+- 2".09, and that of c is ■ — 7=^ = ± 2".0. The corrections in the 

 ~ ' l/l.l 



seventh column are computed by making d' = s, and i = 10', in (12) ; 



the appended probable errors are therefore dr 2".0 zrjri' 



In the probable error of a single comparison is included the probable 

 error of graduation, and that of observation, and the latter is presumably 

 equal to t if the limb and vernier were examined under similar circum- 

 stances. If, therefore, the probable error of one comparison is notably 



