Neill. — Readings of Graduated Circles. 



311 



Equating the two results, and putting B — A = y, we obtain the equation 

 of condition 



y = S + 2e sin (a — 6) . . . . . . (2) 



in which e, S, and 6 are unknown. 



The value of y can be obtained by any two readings of the microscopes, 

 and if four equidistant readings are taken at the points a , a + 90°, 

 a + 180°, and a + 270°, and denoting these values by y . y x . y 2 . y 3 

 respectively, then by putting E = (a — 6) the following four equations 

 are obtained :— 



s + 2e sin E = s + 2e sin E, 

 s + 2e sin (E + 90°) = s + 2e cos E, 

 s + 2e sin (E + 180°) = s-2e sin E, 

 s + 2e sin (E + 270°) = s - 2e cos E, 



y 



1/2 



from which we find 



£ (t/o + t/i + y-2 + ?/: 



tan E 



(3) 

 (4) 

 (5) 



2/0-2/2 

 2/i -2/3 



e = (t/o -y*) cosec E 



In order that the eccentricity may be determined with greater accuracy 

 the circle may be read at a great number of equidistant points. Each pair 

 of readings of the opposite verniers or microscopes furnishes an equation 

 of condition of the form (2), and from all these equations the value of the 

 eccentricity can be deduced by the method of least squares. 



In Chauvenet's Astronomy, vol. 2, three theorems, relating to periodic 

 functions, are given to facilitate the solution of the equations. The result 

 leads to a simple rule, which is as follows : Tabulate the values of y as a 

 distance on a bearing of the reading under the microscope, and reduce these 

 on the meridian and perpendicular as in an ordinary traverse. Then we have 



Zy sina 



(o) 



tan E 



S«/ cosa 



n 2 e 2 = X 2 + Y 2 



(') 



The following values of y B — A were obtained from the horizontal 

 limb of a 5 in. theodolite, by Troughton and Simms, London. The gradu- 

 ations were known, by experience in the field, to be almost perfect. A power- 

 ful micrometer microscope was used to obtain the values of B — A from 

 0° to 360° at intervals of 30°. The readings are in terms of the divisions of 

 the micrometer, and are converted to arc in the second column : — 



0° - 0-014 M.D. or in arc - 23" 



