CHAPTER VI.

ON THE METHOD OF DETERMINING THE AZIMUTH OF THE REFERRING MARK FROM AN OBSERVATION TAKEN TO A CIRCUMPOLAR STAR, AT THE TIME OF ITS MAXIMUM ELONGATION.

WHEN the north polar distance of a star falls short of the latitude of a place, it becomes what is called a circumpolar star at that place, that is to say, a star which never sets, but is continually above the horizon, describing in the course of a siderial day, a small circle round the pole, of greater or less magnitude, according to the length of the star's north polar distance.

In the diurnal circular path, above adverted to, of a circumpolar star, there are two points, one of which is furthest east, and the other furthest west from the north pole. When the star arrives at the one or the other of those points, it is said to be at its maximum elongation, that to the east, being called the eastern, while that on the west, is styled the western elongation.

When a circumpolar star arrives at its maximum elongation, it becomes on account of its slow azimuthal motion a very convenient object for observation for the purpose of determining the azimuth of the referring mark. For this purpose, the elements which are required to be known with reference to it, are three in number, and they are as follows:-1st, the time of the elongation, 2nd, the star's azimuth, and 3rd, its

altitude. The formula whereby these elements may be com

In these expressions, æ stands for the star's north polar distance, and λ for the latitude of the place; P, A, and alt. being the required elements, the first, the star's Horary Angle, the second, its azimuth, and the third, its altitude.

TYPE OF COMPUTATION.

a Ursa Minoris observed at Kaliana, G. T. Station, on the afternoon of the 5th October 1836.

P =89 6 55.56

SinA...

A = 1 47 42.55 Alt. = 29 31 32.43

When the Horary Angle is brought out in the way directed above, it will be in space, and will require to be converted into time. When this reduction is made, the resulting element added to or subtracted from, the star's apparent right ascension, will furnish the siderial time, in the former case of the western, and in the latter of the eastern elongation, which siderial time may be converted to corresponding mean solar or chronometer time, as may be required, agreeably to the precepts given in Chapter V.

EXAMPLE.

Thus the Horary Angle computed above, converted to time, will be 5h 56m 275-70, and the star's apparent right ascension is 1h 1m 42.725; hence the siderial times of the eastern and western elongations are 19h 5m 15o·02 and 6h 58m 10-42, the same in mean solar time being 6h 8m 203, 17h 59m 193.

After the preliminary computation has been gone through, the next step is to take the required observation upon the circumpolar star, which may be done in the following manner About a quarter of an hour before the maximum elongation, plant the theodolite over the station dot, and perform thereon all the necessary adjustments. This done, take a reading to the referring mark. To this reading apply the angle* between the referring mark and the star, the resulting reading is obviously the azimuthal direction of the latter. When the instrument is set in this direction, and the telescope raised to the computed altitude, intersect as accurately as possible the star, which will be found near the cross wires. The maximum elongation not having as yet occurred, the star will be receding from the meridian, continue therefore intersecting it, until it reaches the utmost limit in the direction of its motion. When an intersection has been obtained at that limit, read off the instrument, after which take a second observation to the referring mark.†

The mean between the two readings of the referring mark may be treated as one reading, and the difference between it and the reading of the star will be the angle between the two objects observed, which angle being applied to the star's computed azimuth, will furnish the azimuth of the referring mark.

The observation above mentioned appertains to one of two faces of the zero, to which the instrument was set. On the succeeding night, a similar observation will require to be made

* It will be sufficient if this angle is known to within 2′ or 3′, and a result within this limit may always be obtained in the following manner :Determine an approximate azimuth of the referring mark by an observation upon the sun as explained at pp. 661, 662, add this azimuth to the star's azimuth or subtract one from the other, according as the referring mark and the star lay on different or on the same side of the meridian: the sum or difference so derived, will be the angle sought.

†This process will not be necessary when a good chronometer is at hand, because the time of the maximum Elongation being then accurately indicated ; an observation on the star at that instant, will furnish the angle sought.

on the opposite face of the same zero. In this way all the zeros being disposed of, the mean of all the observations will be the true azimuth of the referring mark.

Whenever practicable, the azimuth of the referring mark ought to be derived from two elongations of a star. When this can be done, the deduced azimuth will not be affected by the errors which may exist in the given latitude and north polar distance. On the other hand, an azimuth obtained from a single elongation will be impregnated with the full effect of those errors.

*

It ought to be mentioned at this place that prior to the year 1832, all the azimuths in the Great Trigonometrical Survey of India were determined by observations taken to stars at their maximum elongations. The method is susceptible of great accuracy, as will appear from an inspection of the following Table extracted from Col. Everest's Indian Arc, published in 1830, and containing a record of the observations made on a Ursa Minoris at the time of its western elongation, together with the azimuths of the referring mark deduced therefrom.

* As the apparent North polar distance of a star is continually changing its value from one day to another, it is clear that the horary angle and the azimuth, which are derived from it, will require a fresh and independent computation, for every elongation observed. This is a tedious process which

may be easily avoided by using the following differential formulæ.

wherein de stands for the variation (supposed to be given in seconds) which has taken place in the star's North polar distance since the 1st day's observation, dP, dA being the corresponding alterations, the former in the horary angle and the latter in the azimuth: compute therefore the azimuth A, the horary angle P, and the altitude of the star for the first elongation observed, and then making do the difference between the North polar distance on the 1st and any subsequent day of observation; deduce Pand dA, and apply them respectively to P and A, the resulting terms will obviously be those which appertain to the star's polar distance + & œ.

As to the signs of dP and dA, it will be remembered that the former will be negative and the latter positive, when the star's North polar distance is increasing; and that they will be of the contrary affections when the North polar distance is diminishing.