Neill. — Readings of Graduated Circles. 315 



To find the amplitude e we have 



2e= - 1 "-07 cos 262° 57' - 8"-65 sin 262° 57' = + 8"-61 

 e = + i"S. 

 The period of the first harmonic is 180°. 

 Again y 2 = 2"*91 cos 2a — 0"71 sin 2a. 

 When the function vanishes ; 2a becomes E' ; 



tan Jii = q^j 



E' = 76° 17' 

 and 2e' = 2""91 cos 166° 17' - 0"'71 sin 166° 17' = - 3"-00 

 e' = - l"-50. 



The period of the second harmonic is 90°. The coefficients of the third 

 and higher harmonics are so small that they may be dismissed as due to 

 accidental errors of reading. 



The observed value of B — A is therefore represented by 

 B _ A = - 24"-25 + 8"-61 sin (a - 173°) - 3"-00 sin (2a - 76°) . . (9). 



The first term is the difference of the angular distance between the 

 verniers and 180°. 



The second term, 8"" 61 sin (a — 173°), is, properly speaking, the re- 

 sultant of any errors of centring, due to two principal causes: First, any 

 error in centring the circle with the dividing-engine when the main divi- 

 sions were engraved ; second, any error in centring the circle with the axis 

 of the alidade. 



Any one of these causes, or a combination of them, provided they do 

 not cancel each other, will be eliminated by reading two opposite verniers 

 and taking the mean of the readings. 



The third term seems to be too large to be dismissed as due to accidental 

 errors of reading. 



Since the period of the harmonic is 90°, we may be justified in assuming 

 that four main divisions of the circle 90° apart were directly copied from 

 the dividing-engine and the smaller divisions on each quadrant performed 

 automatically. 



On this assumption we see that errors due to the third term are elimi- 

 nated by verniers placed 90° apart and the mean of their readings taken. 



From these results we can deduce a method of shifting the verniers 

 around the circle when a number of readings are taken to obtain a mean 

 result. 



Four readings taken 90° apart will eliminate all the periodic errors. 

 The programme is : — 



First round — Set vernier A on a. 

 Second ,, Set vernier A on a + 90°. 

 Third „ Set vernier A on a + 90° + 45°. 

 Fourth „ Set vernier A on a + 135° + 90°. 



In the event of a greater number of readings being taken, the third 

 term is eliminated by any multiple of four, so that eight settings 45° apart 

 should be made. 



The vernier A is shifted 45° for the first four settings, then 22^° for the 

 fifth, and 45° for the remaining settings. 



This method is frequently used on triangulation surveys in the Dominion. 



The equation (9) may be tabulated and a comparison made with the 

 observed values from which the mean error of a reading of two opposite 

 verniers can be obtained. 



