310 



Transactions. 



The eccentricity is of more importance in surveying, when the angles of 

 a traverse are obtained by reading one of the verniers, than in astronomical 

 and geodetic observations, when both microscopes are invariably read. 

 Sufficient care is usually taken by the makers in adjusting the centres to 



x X 



Fig. 1. Fig. 2. 



reduce the eccentricity to a very small amount. An error of i^Vo °f an 

 inch in the centring of a 5 in. theodolite will cause a maximum error of 

 V 22" in the reading of a bearing when it falls on that part of the circle 

 where the effect of the eccentricity is greatest. 



To find the eccentricity : In the diagram, fig. 2, let 



a denote the microscope or vernier reading. 



x = true reading. 



= XD. 



e = eccentricity = OC. 

 C is the centre of the circle, and the centre of the alidade. 

 When the microscope reading is at A or B the true reading is at a or b. 

 Since e is small, the arc Aa = a — x is sensibly equal to the perpendicular 



PC from C on AB. 

 Now PC = e sin A o D, or a — x = e sin (a — 0) . . . . (1) 



When a - 6 = 90° or 270°, e = ± (a - x), 

 and for a - 6 = 0° or 180°, e = 0. 



Again we have for the opposite microscope 



b - x = e sin (180° + a - 6) = - e sin (a - 6). 

 Equating this with (1 ), 2x = a+ b. 



x = \ (a + b). 



The eccentricity is therefore a periodic function which is eliminated by 

 taking the mean of the readings of two opposite microscopes, or more gene- 

 rally by any number of equidistant microscopes. In accordance with this 

 principle, circles are usually equipped with the zeros of the microscopes 

 nearly 180° apart ; but as they may not be perfectly adjusted at the dis- 

 tance of 180° we shall put 



180° + S = the angular distance of the microscope B from A. 

 Then if (a) is the division under microscope A, A and B the readings of the 

 two microscopes, the true reading for the microscope A is 



x = A + a + e sin (a — 6), 

 and for the microscope B 



180° + S + x = 180 + B + a + e sin (180 + a - 6), 



or x = B — S + a — e sin (a — 0). 





