CHAPTER VII. ON THE METHOD OF DETERMINING THE AZIMUTH OF THE REFERRING MARK, FROM OBSERVATIONS TAKEN TO A CIRCUMPOLAR STAR NEAR THE TIME OF ITS MAXIMUM ELONGATION, AS PRACTISED IN THE GREAT TRIGONOMETRICAL SURVEY OF INDIA, ALSO ON THE MODE OF FINDING THE VARIATION OF THE NEEDLE. It is clear that when a circumpolar star is taken in the way described in the preceding chapter, only one observation can be made at one elongation. This is a great limitation of the powers of the observer. To extend this power, the procedure, introduced by Col. Everest into the Trigonometrical Survey, consists in taking a circumpolar star, a certain number of times before and after its greatest elongation; and in subsequently reducing these observations to the star's maximum position, and then working out the azimuth as before explained. In addition to a theodolite, a good chronometer is absolutely necessary to carry this process into effect. Col. Everest's method of taking a circumpolar star may be described as follows:-About an hour before the maximum elongation of a star selected for observation, the observer will adjust the theodolite over the station dot, and set it to a given zero. When this is effected, take a reading to the Referring Mark; and then having fixed the telescope to the star's computed altitude, move it azimuthally by the hand, until the star appears in the field of vision. Now fasten the horizontal clamp, and by the usual appendages of slow motion, place the star in the upper angle of the wires, if it is descending, or in the lower angle, if it is ascending. This done, call out to the assistant to count the seconds' beats of the chronometer, at the same time watch the star's approach to the intersection of the wires. As soon as the star comes over the said intersection,* mark the time, and then read off the azimuthal limb. Now loosen the horizontal clamp, and after moving the telescope by the hand a few degrees in advance, bring it back to the star, and then take another intersection thereon in the same way as before; after which lower the telescope and make an observation on the Referring Mark. This will complete a set of observations on one face of the theodolite. As to the manner of treating these observations, it will perhaps be useful to note that one angle will be derived from the first pair of the readings of the Referring Mark and the star, while another angle will be obtained from the second pair of the readings taken on the same objects. When observations on one face of the instrument have been made as described above, the observer will now reverse the face of the theodolite, and take a second set of intersections similar to the first. In this manner, when he has done with one face, he will revert to the other, until, as may required, four or six changes of face are regularly gone through. This will complete observations on one zero, at a given elongation of a circumpolar star. be The system of changing the zero of a theodolite as explained at pp. 403-405 will require to be practised in circumpolar star observations in the same rigorous manner as in observations on terrestrial signals; for the graduation errors which that procedure is supposed to correct, have a tendency to vitiate equally the two classes of observations, and in both cases, therefore, they must be eliminated by similar arrangements and artifices. When a circumpolar star is being observed, it is convenient to adjust the changes of zero bythe Referring Mark. *To do this well, will require a little practice. After two or three trials, the observer will know the direction of the star's motion, and when he has acquired this, he will intuitively place the cross-wires, so that the star may at once come upon it. SPECIMEN OF THE ANGLE BOOK. a Ursa Minoris at Eastern Elongation observed at Kaliana, G. T. Station, on the afternoon of the 5th October 1836, with a three feet Theodolite. ing circumpolar star observations. The subjoined is a specimen of the Angle Book for register To obtain the best angles which a theodolite is capable of furnishing, the motion of the telescope, whether proceeding from the Referring Mark to the star, or vice versa, should be continuous and in one direction, never allowing the telescope or rather the cross-wires contained therein, to pass the object to be intersected, and then be brought back to it. This mode of taking an observation, although difficult at first, is rendered very easy after a little practice. With a view of computing these observations, the first thing to be done is the conversion of the observed chronometer times to corresponding siderial times, the mode of executing which has been explained in Chapter V. When this reduction is made, take the difference between the siderial time of each observation and that of the maximum elongation, and convert it into space. Let SP stand for the elements so derived. Now &P being the interval elapsed between each observation and the star's maximum position, the term, which is now required to be known, is the azimuthal variation A corresponding thereto. This term may be computed by the following formulæ : 1st. When the star is observed below the maximum position. 2 sin2dP tan &A 2nd. = sinP cot œ cos \ {1+ tan3æ cos dP+ sec2œ cotP sin dP } When the star is taken above the maximum position. 2 sin2dP sinP cot œ cos \{1+ tan3œ cos §P— sec2œ cotP sin dP } in which as stated elsewhere A represents the latitude of the place, a standing for the star's North polar distance, and P for the horary angle at its maximum position East or West.* * As the star ascends on the East side of the meridian, the observations made before the Eastern elongation are reduced by the first formula, and those taken after, are computed by the second. In the Western elongation, a contrary procedure is followed, because the star is descending; the second formula being used in deducing the prior, and the first in computing the subsequent observations. These formulæ have been investigated by Babu Radhanath Sickdhar, chief computer to the Great Trigonometrical Survey, and are applicable to all circumpolar stars, irrespective of the lengths of their North polar distances, and they are now used in all the rigorous computations of the Great Trigonometrical Survey. The terms A being computed and applied to the observed angles, we obtain the angles as if taken at the star's maximum elongation. To these angles, the star's computed azimuth being applied, the resulting elements will be the required azimuths of the Referring Mark. It will be seen that this deductive process, although suited to the requirements of a Trigonometrical Survey, will prove much too operose, if applied to an operation of a lower order. To meet the wants of the latter, therefore, we will describe an approximate method of computation, derivable from the above formulæ, and which when applied to a Ursa Minoris, will not produce an error of a second in the result. This approximate process of computation is as follows: 1st. Compute the following constant logarithm. 0.29303+ log sec + log tan a + log. cosec P. 2nd. Compute as accurately as the means will allow to the nearest second, the chronometer time of the star's maximum elongation observed. 3rd. Compute the chronometer interval elapsed between each observation and the maximum elongation, and convert it to minutes and decimals thereof. 4th. Take the logarithm of the interval converted to minutes as directed above, double it, and add thereto the constant log, deduced according to precept 1st. The natural number answering to the sum is 4 in seconds. 5th. In making this computation, the logarithms used need not be carried beyond 5 decimals. To carry this method into effect, we would recommend to the Surveyor to derive his azimuths from observations made to a |
