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38 36 15.6
90 0 0
51 23 44.4
When the sun is the object observed, the hour angle P (as in the last example) is the apparent time from apparent noon at the place of observation, converted into space; but with a star, it is its distance from the meridian, either to the east or west, according as it has or has not come to its culmination; and this angle is simply the sum or difference of the star's right ascension, and the time of the observation converted into sidereal time; to be multiplied by 15 for its conversion into space.
Method 5th.-By two observed altitudes of the sun, or a star, and the interval of time between the observations.
This problem is of importance, as its solution, though long, does not involve a knowledge of the correct time at the place of observation; and the short interval of time can always be measured with sufficient accuracy by any tolerable watch. Various methods have been devised to shorten the calculation of "double altitudes" by tables formed for the purpose, one of which may be found at page
231 of Riddle's "Navigation;" but the direct method by spherical trigonometry is most readily understood and easily followed.
Let S and S' represent the places of the object at the times of the two several observations, (and they may be on different sides of the meridian, or, as in the figure, both on the same side); ZS and ZS' then are their respective zenith distances, and PS and PS'
their polar distances; SPS' being the hour angle observed.
First-In the triangle PSS', the two sides PS and PS' are given, with the included angle at P to find SS and the angle PSS. Again, in the triangle ZSS', we have the three sides to find the angle ZSS', which, taken from PSS' just found, leaves the remaining angle PSZ. Lastly—in the triangle PSZ we have PS, ZS, and the angle PS Z, to find PZ, the co-latitude sought. In a similar manner the latitude may be found by simultaneous altitudes of different stars, the difference of their right ascensions giving the angle SPS, without the use of a watch. Tables have been calculated by Dr. Brinkley, from which the distance SS can be obtained by inspection (allowing for the change in the right ascension of the stars after any long interval), and the calculation is thus considerably abridged. By an azimuth and altitude instrument, the latitude may also be found by the two altitudes, and the difference or sum of the observed azimuths of the sun or star.
Equal altitudes of the same star on different sides of the meridian, with the interval of sidereal time, between the observations, also furnish the means of ascertaining the latitude, and this method is most useful in a perfectly unknown country. The hour angle, east or west, will evidently be measured by half the elapsed interval of time; and in the triangle Z PS, we have this hour angle ZP S, the polar distance PS, and the coaltitude Z S, to find ZP the co-latitude; moreover, the hour angle being known, and also the right ascension of the star, the point of the equinoctial which is on the meridian, and consequently the
local sidereal time is determined, from which the mean time can be deduced.
The latitude may likewise be ascertained, independently of the graduation of the instrument, by placing the axis of the telescope of an altitude and azimuth circle* due north and south, so that the vertical circle shall stand east and west. The observations of the two moments T and T' (in sidereal time), in which the star passes the wire of the telescope, will give the latitude from the following formula.
Cot L cot declination x cos
If a chronometer set to mean time is used, the interval (T—T') must be multiplied by 1·0027379, or the value corresponding to the interval, found in the table for converting mean into sidereal time, must be added †.
The accuracy of this method depends upon the correctness of the tabulated declination of the star, but a slight error in this will not affect the difference of latitude between two places, thus found. By observing on following days with the axis reversed, and taking the mean of the observations, any error in the instrument is corrected; this method is particularly recommended by Mr. Baily for adoption in geodesical operations, as the difference of latitude of two stations is obtained almost independently of the declination of the star, and the only material precaution to be taken is in levelling the axis of the telescope, which should be one of very good quality.
TO FIND THE TIME.
Method 1st.-From single, or absolute, altitudes of the sun, or a star whose declination is known, as also the latitude of the place.
This problem is solved by finding the value of the horary angle P, in the same "astronomical triangle" Z PS, whose elements have
* A portable transit placed in the plane of the prime vertical, instead of that of the meridian, of course affords the same facility for thus determining the latitude. The stars selected should have their declinations less than the latitude of the place, but by as small a quantity as possible.
Table 7. Baily's Astronomical Tables and Formulæ.
already been described. In this case, the three sides, viz., the colatitude, the zenith, and polar distances, are given to find the hour angle P, which, when the sun is the object observed, will (as was explained in page 193) be the apparent time from apparent noon at the place of observation; and it is converted into mean time by applying to it the equation of time with its proper sign. In the case of a star, it will denote its distance in time from the meridian, which being added to its right ascension, if the observation be made to the westward of the meridian, or subtracted from the right ascension (increase by 24 hours if necessary) if to the eastward, will give the sidereal time, to be converted into mean solar time, if required.
A simple formula for finding the angle of a spherical triangle whose three sides are given is sin. 2x= sin (Sc) (sin S-b) sin c. sin b where S denotes the sum of the three sides a, b, and c; of which a is assumed as the one opposite the required angle. In the present case a represents the co-altitude or zenith distance; b the co-declination, or polar distance; and c the co-latitude.
Observed altitude of the upper limb of the sun on May 4,
Required the error of the watch.
14° 44′ 58′′
0 0 28
Zenith distance (ZS)
Semidiameter at 6.75h.
Apparent altitude e
* The most favourable time for observing single, or absolute, altitudes of the sun or a star, to obtain the local time, is when they are on or near the prime vertical, since their motion in altitude is then most rapid, and a slight error in the assumed latitude is not of so much consequence. The corrections for the refraction, however, are then often considerable. The same observation will of course give the azimuth Z, and also the variation of the needle, if the magnetic bearing of the star, or of either limb of the sun, is taken by another observer at the same moment as the altitude. This will be further explained.