Art. XLVI.—Methods of Observing to eliminate the Periodic Errors affecting the Readings of the Graduated Circles in Astronomical and Surveying Instruments.

Transactions and Proceedings of the Royal Society of New Zealand, Volume 46, 1913, Page 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 consideration 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 are represents on a 3 in. theodolite, an angle which this instrument is capable of measuring accurately. We find the fraction to be nearly the 1/2300 part of 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 division 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 dividingengine, 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 O 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 are 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 Bb equal and parallel to Cc, when A and B are two reading microscopes diametrically opposite. Resolving the small spaces Aa and Bb 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 are too great and the other an are 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 microscopes and taking the mean of the readings the error due to eccentricity is eliminated.

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 reduce the eccentricity to a very small amount. An error of 1/1000 of an inch in the centring of a 5 in. theodolite will cause a maximum error of 1′ 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. θ = XD. e = eccentricity = OC. C is the centre of the circle, and O 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 are 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 − θ) (1) When a − θ = 90° or 270°, e = ± (a − x), and for a − θ = 0° or 180°, e = 0. Again we have for the opposite microscope b − x = e sin (180° + a − θ) = − e sin (a − θ). Equating this with (1), 2x = a + b. x = ½ (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 generally 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 distance 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 − θ), and for the microscope B 180° + S + x = 180 + B + a + e sin (180 + a − θ), or x = B − S + a − e sin (a − θ).

Equating the two results, and putting B − A = y, we obtain the equation, of condition y = S + 2e sin (a − θ) (2) in which e, S, and θ 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 a0, a0 + 90°, a0 + 180°, and a0 + 270°, and denoting these values by y0, y1, y2, y3 respectively, then by putting E = (a0 − θ) the following four equations are obtained:— y0 = s + 2e sin E = s + 2e sin E, y1 = s + 2e sin (E + 90°) = s + 2e cos E, y2 = s + 2e sin (E + 180°) = s − 2e sin E, y3 = s + 2e sin (E + 270°) = s − 2e cos E, from which we find s = ¼ (y0 + y1 + y2 + y3 (3) tan E = y0 − y2/y1−y3 (4) e = (y0 − y2) cosec E (5) 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 tan E=σy sina/σy cosa (6) n2e2 = X2 + Y2 (7) The following values of y B−A were obtained from the horizontal limb of a 5 in. theodolite, by Troughton and Simms, London. The graduations were known, by experience in the field, to be almost perfect. A powerful 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 are in the second column:— 0° - 0·014 M.D. or in arc - 23″ 30° - 0·018 " - 29″ 60° - 0·020 " - 33″ 90° - 0·022 " - 36″ 120° - 0·020 " - 33″ 150° - 0·015 " - 25″ 180° - 0·013 " - 21″ 210° - 0·010 " - 16″ 240° - 0·012 " - 20″ 270° - 0·012 " - 20″ 300° - 0·009 " - 15″ 330° - 0·012 " - 20″

Tabulating these values and resolving each into two, one in the direction of the meridian and the other at right angles to it, we obtain the following:— Form A. On Meridian. On Perpendicular. a. y. + − + − 0° −23″ 23·0 30° −29″ 25·1 14·5 60° −33″ 16·5 28·6 90° −36″ 36·0 120° −33″ 16·5 28·6 150° −25″ 21·7 12·5 180° −21″ 21·0 210° −16″ 13·9 8·0 240° −20° 10·0 17·3 270° −20″ 20·0 300° −15″ 7·5 13·0 330° −20″ 17·3 10·0 Sums −291″ 83·1 89·4 68·3 120·2 83·1 68·3 −6·3 −51·9 12e sin E = − 6·3 − log = 0·79934 12e cos E = − 51·9 − log = 1·71517 E = 186° 55′ + log tan E 9·08417 e = + 4″.3 + log12e = 1·71864 s = 1/12 (−291)= −24″ A single reading of the vernier A requires a correction for eccentricity of + 4″.3 sin (a − 186° 55′) The distance between the verniers is 180° − 24″ = 179° 59′ 36″. Having found the error in the readings due to eccentricity and the constant angle between the two verniers, we may correct the observed readings for these errors, and the residuals will be the errors due to the graduation of the circle and accidental errors of reading. A circle may be graduated by copying a certain number of the main divisions directly from the dividing-machine, and then the smaller divisions between these points are performed automatically by setting the machine in motion. If the circle is not truly centred with the dividing-engine there will be an error in the main divisions due to the eccentricity and an error in the divisions of the smaller ares, with a shorter period, depending on the number of main division directly copied from the dividing-engine. These may be investigated in the same manner as for the centres of the circle and the alidade not coinciding, or they may be determined by the method of harmonic analysis as follows by putting y = A0 + A1 cos a + A2 cos 2a + &c. + B1 sin a + B2 sin 2a + &c. (8)

Before proceeding with the analysis we have to consider the two systems of measuring angles. Graduated circles are usually numbered from 0° to 360°, the numbers increasing in the clockwise direction, and azimuths or bearings are referred to the north point as zero. The four cardinal points correspond to North, 0° South, 180° East, 90° West, 270° This may be termed the practical method of angular measurement. In trigonometry the angles are referred to the east point as zero and increase counter-clockwise. The four cardinal points are represented by North, 90° South, 270° East, 0° West, 180° If we call this the theoretical method of measuring angles, and if we bring the two zero points to coincide, an angle is converted from one system to the other in this instance by a, theoretical = 360° − a, practical. Applying this conversion formula to the value of E = 186° 55′, we obtain E (as an azimuth) = 360° − 186° 55′ = 173° 05′, and the line of no eccentricity intersects the graduated circle at the points 173° 05′ and 353° 05′. Hence the reading of the vernier A requires a correction of + 4″.3 sin (a − 173° 05′) Taking the observed values of B—A and applying the correction for the angular distance between the two verniers, we obtain the errors due to eccentricity and imperfect graduation. Then computing the amount of eccentricity, the residuals represent the errors due to imperfect graduation and accidental errors of reading. The following form is a convenient method of tabulating these results:— Form B. (1.) (2.) (3.) (4.) (5.) (6.) (7.) (8.) a. B−A (2) + 24″.25 8″.6 x sin (a − 173.). −3″.0 x sin (2a − 76.). Computed B − A. Errors of Graduation and Reading. a. ° ″ ″ ″ ″ ″ ″ o 9 − 23 + 1·25 − 1·06 + 2·91 − 22·4 − 0·6 0 30 − 29 − 4·75 − 5·18 + 0·83 − 28·6 − 0·4 30 60 − 33 − 8·76 − 7·92 − 2·08 − 34·2 + 1·2 60 90 − 36 − 11·75 − 8·54 − 2·91 − 36·7 − 0·3 90 120 − 33 − 8·76 − 6·87 − 0·83 − 32·0 − 1·0 120 150 − 25 − 0·75 − 3·36 + 2·08 − 26·5 + 0·5 150 180 − 21 + 3·25 + 1·05 + 2·91 − 20·3 − 0·7 180 210 − 16 + 8·25 + 5·18 + 0·83 − 18·2 + 2·2 210. 240 − 20 + 4·26 + 7·92 − 2·08 − 18·3 −1·7 240 270 − 20 + 4·25 + 8·64 − 2·91 − 18·6 − 1·4 270 300 − 15 + 9·26 + 6·87 − 0·83 − 19·2 + 4·2 300 330 − 20 + 4·25 + 3·36 + 2·08 − 18·8 − 1·2 330 As a check on the results in column (4) the values of B − A can be analysed by the formula (8)— y = A0 + A1 cos a + A2 cos 2a + &c. + B1 sin a + B2 sin 2a + &c.

It will generally suffice to compute the first and second coefficients A1 and A2, and B3, which can readily be done by the method shown on form A. A scheme proposed by Professor Runge (“Zeitschrift für Mathematik und Physic”) is given in Gibson's “Calculus,” in which all the coefficients are easily computed. The following is Runge's scheme slightly altered:— Harmonic Analysis of Column (2). 0. 1. 2. 3. – − 2·0 − 8·0 + 5·0 C' diff. − 44·0 − 90·0 − 101·0 − 56·0 C sum. − 21·0 − 41·0 − 53·0 − 23·0 − 49·0 − 48·0 − 56·0 − 53·0 − 41·0 2·0 − 23·0 − 29·0 − 33·0 − 36·0 − 33·0 − 25·0 − 21·0 20·0 − 16·0 − 20·0 − 20·0 − 16·0 − 9·0 − 18·0 − 16·0 − 13·0 − 9·0 − 9·0 − 13·0 − 18·0 − 31·0 − 16·0 d sum. 0·0 − 5·0 d' diff. 1 2 3 Cosine Terms. Sine Terms. 0 and 6 1 and 5. 2 and 4. 3 1 and 5 2 and 4 3. − 44·0 − 90·0 − 2·0 − 6·9 − 44·0 45·0 − 2·0 − 9·0 − 26·9 0·0 − 4·3 − 18·0 + 101·0 − 56·0 − 2·5 + 50·5 + 56·0 − 6·0 − 16·0 + 16·0 − 145·0 + 0·5 + 6·5 − 7·0 − 25·0 0·0 − 2·0 − 146·0 − 6·9 + 11·0 − 26·9 + 4·3 − 291·0 − 6·4 + 17·5 − 51·9 − 4·3 + 1·0 + 7·4 − 4·5 + 1·9 + 4·3 A0 = − 24·25 A1 = − 1·07 A2 = 2·91 A3 B1 = − 8·65 B2 = − 0·71 B3 A6 = + 0·08 A5 = + 1·23 A4 = − 0·75 − 1·17 B5 = + 0·31 B4 =+ 0·71 − 0·33 Denoting by y1, y2, &c., the first, second, &c., harmonics, the equation to the curve is y = − 24″.25 + y1 + y2 + &c. or y + 24″.25 = y1 + y2 + &c., by changing the origin. Now y1 = − 1″.07 cos a − 8″.65 sin a. The value of (a) wten y1 vanishes is E, therefore − 1″.07 cos E − 8″.65 sin E = 0. tan E= − 1·07/8·65 E = 172° 57′.

To find the amplitude e we have 2e = − 1″.07 cos 262° 57′ − 8″.65 sin 262° 57′ = + 8″.61 e = + 4″.3. The period of the first harmonic is 180°. Again y2 = 2″.91 cos 2a − 0″.71 sin 2a. When the function vanishes; 2a becomes E'; ∴ tan. E' = − 2·91/0·71 E' = 76·17′ and 2e' = 2″.91 cos 166° 17′ − 0″.71 sin 166° 17′ = − 3″.00 e'= − 1″.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 resultant of any errors of centring, due to two principal causes: First, any error in centring the circle with the dividing-engine when the main divisions 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 eliminated 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 22½° 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.

In form B the fourth column contains the values of + 8″.6 sin (a − 173°). The fifth column contains the values of − 3″.0 sin (2a − 76°). The sixth column is the computed values of B − A = − 24″.25 + 8″.6 sin (a − 173°) − 3″.0 sin (2a − 76°). Subtracting these from the observed values of B − A in column (2) we obtain the residuals in column (7), which represent the accidental errors of graduation and reading. These results show that the circle is exceedingly well graduated. The accidental errors of graduation are not reducible to any regular law. They may occur at any division of the circle with either positive or negative signs with equal probability. They may be found directly by testing the divisions with a micrometer microscope. On an instrument equipped with verniers, the angle between any two divisions of the vernier may be used for this purpose. Such errors may be reduced by a greater number of reading microscopes or verniers, but they cannot be wholly eliminated by any special method of observation. Fig. 3. A graphical representation of the results is shown in fig. 3. The firm curve shows the values of B−A as ordinate, for corresponding values of x, 0°, 30°, &c. The two dotted curves are the first and second harmonics.