OF ECCENTRICITY AND GRADUATION. 7 



If we had access to the mean arc, equation (2) would enable us to find 

 the correction for any angle whatever by making three comparisons with 

 the standard circle, the index being set successively at 0°, and at any two 

 other points. For the differences between the reading of the circle with 

 the index at 0° and each of the other two readings would furnish two 

 values of;', in which the error of observation could be diminished by repe- 

 tition to any extent desired. By substituting these values of y and the 

 corresponding ones of / in (2), two equations would be obtained deter- 

 mining A and B, which substituted in the same formula (2) would afford 

 an equation giving the correction y — / for any value of /. From the 

 definition of the mean arc it follows that the same result would be ob- 

 tained from readings made with the index set successively upon every 

 line of the actual graduation. Let R be the circle reading corresponding 

 to any setting S of the index, and Z an assumed value of the reading 

 Avhen the index is at 0° of the mean arc, the true reading being Z -f- X, 

 in which Xis unknown. Then, disregarding in each case, for the reason 

 just given, the deviation of the setting from the corresponding position 

 on the mean arc, / = S, and y ^ R — {Z -\- J?).* 



Substituting these values in (2), and making 



• R-8- Z= D, * (3) 



each observation will furnish an equation of the form : 



A %\n h S 4- B \l — co^ h S) + X = D. (4) 



If M is the number of observations, the normal equations will be : 

 A [sin^ h S'\+B [(1-cos i S) sin i S^+Xlsin i S] — [D sin J S]=0^ 

 A [sin i *S (1 - cos i ,S')] -f ^ [(1 - cos * ^)^] -f X[l -cos h S] [ 



-[D (1-cos i 5)]=0 p''^'* 

 A [sin i 8] + ^ [1 - cos ^ S~\ -f MX - [Z>] =0 j 



After substituting the numerical values of the known quantities in 

 these equations, and finding the values of A, B, and X, the correction 

 for eccentricity of any observed reading will be given by (2). From (4) 

 D may be obtained for any value of S, and the difference between this 

 computed quantity and the observed value of D, for each comparison, 

 is the local correction of the graduation, which, however, includes an un- 

 known, and perhaps relatively large, error of observation. 



An examination like that just described, embracing every line of the 

 graduation, and repeated until the effect of errors of observation is suf- 

 ficiently diminished, would afford a complete knowledge of the condition 

 and capabilities of the instrument. For the corrections due to the position 

 of the axis having been obtained, the local correction for each line would 



*The readings of the circle are supposed to increase as the angles indicated by 

 the sextant increase, which is actually the case in the apparatus referred to. 



