Neill. — Readings of Graduated Circles. 309 



Art. XLVI. — Methods of Observing to eliminate the Periodic Errors affecting 

 the Readings of the Graduated Circles in Astronomical and Surveying 

 Instruments. 



By W. T. Neill, New Zealand Survey Department. 



[Read before the Astronomical Branch, Otago Institute, 23rd September, 1913.] 



The correctness of the divisions on graduated circles is an important con- 

 sideration when precise measurements are to be made with them. 



In ancient times an error of 8' or 10' in an astronomical observation was 

 considered negligible, on account of the instruments employed in those 

 days being so imperfect, and therefore incapable of giving more accurate 

 results. At the present time an error of 8" or 10" in such observations is 

 enormous, from which we may form an idea of the great improvements 

 effected in the precision of modern instruments. 



As an example of the minuteness of the quantity measured, we may 

 compute the fraction of an inch which 1' of arc represents on a 3 in. theo- 

 dolite, an angle which this instrument is capable of measuring accurately. 



We find the fraction to be nearly the 23^0 P ar * °f an inch, and to see it 

 as a linear distance the aid of a powerful magnifying-glass is required. 



Formerly the circles for angular measurements were divided by hand, 

 and the errors of the graduations were not easily reducible to law. 



The invention of the automatic dividing-engine by Ramsden, of London, 

 in 1768 was an event of great importance in the progress of accurate divi- 

 sion of the circles intended for angular measurement ; but the graduations 

 made by the engine cannot be assumed to be faultless, and an examination 

 of each circle to determine the errors of graduation and those introduced 

 by imperfections in the instrument is one of the most laborious tasks that 

 the astronomer has to perform, and their elimination by special methods 

 of observation is a matter of first importance. 



The circles on modern instruments are usually graduated by a dividing- 

 engine, and the errors in the divisions are reducible to some law, which can 

 be discovered by an examination of the circles and an analysis of the results, 



The centre of the circle is the centre of the divided rim, but since it 

 turns on an axis which may not be, and commonly is not, coincident with 

 this centre, an error due to eccentricity is introduced. 



To make this clear, let C, fig. 1, be the centre of the circle, and the 

 centre of rotation. Join OC and produce it to XX'. If the circle rotate 

 through an angle XOD, the centre C will describe the small arc Cc. It is 

 plain that the points of the circle which would, if it were accurately centred, 

 have come under the reading microscopes at A and B will have come to 

 the points a and b, found by drawing Aa and B& equal and parallel to Cc, 

 when A and B are two reading microscopes diametrically opposite. 



Resolving the small spaces Aa and B& into two, one in the direction of 

 the limb of the circle and the other at right angles to it, the effect of the 

 first resolved part will be that one microscope will read an arc too great 

 and the other an arc too small by the same quantity than they would have 

 read if the circle were centrically placed. The second resolved part will 

 carry the point at A farther from the microscope, and the point at B nearer 

 to the opposite microscope by an equal distance ; therefore the mean of 

 the readings will remain unaltered. Hence, by reading two opposite micro- 

 scopes and taking the mean of the readings the error due to eccentricity is 

 eliminated. 



