OF ECCENTRICITY AND GRADUATION. 17 



Different series of comparisons, therefore, cannot be expected to agree 

 very closely in the vahies of A they furnish, still less in their values of 

 B, but such a relation should nevertheless subsist among these apparently 

 discordant results as to render their respective sets of eccentric corrections 

 reasonably harmonious. The constants A and J3 are, in fact, as indicated 

 by (1), rectangular coordinates of the index-axis referred to the system 

 in which the center of the sextant arc is the origin, the radius passing 

 through mean 0° is the axis of J., and 1" of the arc having a radius equal 

 to half the length of the index-bar, from axis to edge of vernier, is the 

 linear unit. By substituting the value of Xobtained from the last formula 

 of (6) in the other two, they become the equations in A and B of two right 

 lines, inclined to the axis of A at angles of 121° 35', and 124° 44', respect- 

 ively, and determining by their intersection the position of the index-axis. 

 These lines meet at an angle of but 3° 9', and since errors of observa- 

 tion can only displace them laterally, without altering their direction,^ 

 it follows that, as might be expected, a straight line passing through the 

 index-axis, and perpendicular to the radius through the middle of the arc, 

 is located much more accurately than the position of the axis upon that 

 line. 



Let D' be the observed value of D obtained by any one setting S' of 

 the sextant, all the other values of *S and D which furnished equations of 

 condition retaining those designations ; the local correction of the gradua- 

 tion at S' will then be the difference between D' and the value of D com- 

 puted by (4) for S = S\ or : 



i)' - (a sin iS' + B(l- cos i S') + x\ , 



which, by substituting the values of A, B, and X, in (7), becomes : 



1.5671 sin \ S' — 1.8384 (1-cos * S') — 0.4802 + l) D' + 



- 7.6467 sin J S' -f 11.0275 (1 cos \ S') + 1.567l) D'sin h S' + 

 11.0275 sin J /S" - 17.9311 (1 - cos i S') - 1.8384^ D' (1 - cos h 8')+ 

 1.5671 sin h S' - 1.8384 (1 - cos I S') - 0.4802^ "^ [^ 1 "*" 



- 7.6467 sin A *S' + 11.0275 (1 - cos i/S')-f 1.5671 ) ^ fjDsin h S^ + 

 11.0275 sin iS' — 17.9311 (1 - cos i S') - 1.8384^2'ri)(l-cos i>S)~| . 



*The probable displacement of the line represented by the normal equation in 

 A, which is the one making an angle of 121° SS' with the radius through 0°, is t 

 X 0.79 ; the probable displacement of the other line is ^ X 0.92. 



2 



