An observer not furnished with a mural circle, or other instrument fixed in the plane of the meridian with which to measure meridional altitudes, can obtain his latitude more correctly than by observing a single approximate meridional altitude with a sextant or other reflecting instrument, by taking a number of altitudes of the sun or a star near to, or on each side of the meridian, and from thence determining the correct altitude of the object at the time of its culmination. This method, termed that of "circum-meridian altitudes," to the mean of which altitudes is to be applied a correction for its “ reduction to the meridian," is susceptible of great accuracy; and the repeating circle, already described, is peculiarly adapted for these observations, on account of the rapidity with which they can be taken. The distance of the sun or star from the meridian (in time) is noted at the moment of each observation, by a chronometer when the former is the object, and by a sidereal clock (if there is one) when the latter, to save the conversion of one denomination of time into the other. The formula given by Mr. Baily, freed from the second part of the equation which it is seldom necessary to notice, is— x represents the required correction in seconds. L, the latitude (known approximately). D, the declination (minus when south). Z, the meridional zenith distance, also known approximately from the above. A, a quantity depending upon the horizontal angle of the object, and given in the 13th table, page 240, under the head of “Reduc tion to the meridian," being = 2 sin2 P where P the horary angle at the pole, as shown by a well-regulated clock; which angle will change its sign after the meridional passage of the star. Among the instructions drawn up by Mr. Airy for the guidance of the officers employed upon the survey of the North American Boundary, this method of determining the latitude with the altitude and azimuth instrument is recommended, and was constantly practised with stars near the meridian. The axis of the instrument is to be adjusted nearly vertical, and the cross axis nearly horizontal (great accuracy is not required), the telescope made to bisect the star upon its middle horizontal wire, and the time noted. Then read the large divisions with the pointer, and the two microscopes A and B; read also the level right hand and left hand. Turn the instrument 180° in azimuth, and repeat these observations-revert to the first position, and continue this process as often as may be thought necessary-note the barometer and thermometer-then add together Reading of A. And equivalent for left-hand level. Subtract equivalent for right-hand level. Divide the remainder by 2, and apply the pointer reading of A for the uncorrected circle reading for the first observation. The same process is repeated for the second and all the other observations. For each observation correct the chronometer time for rate and error, and convert this into (if not already showing) sidereal time ; take the difference between the sidereal time and the star's right ascension for the star's hour angle, which reduce to seconds of time and call p. Then compute for each observation the number which is the correction in seconds of arc to the observed zenith distance to bring it to the true meridian zenith distance, and is always subtractive, except the star is below the pole. In applying this correction however, to the circle readings, it will be additive, or subtractive, according as, by the construction of the circle, increasing readings represent increasing or decreasing zenith dis tances. Half the difference of two corrected readings in opposite positions of the instrument is the star's apparent zenith distance on the meridian; or the mean of all the observations in one position may be compared with the mean of all those in the other, and half their sum is the zenith point. To this zenith distance add the correction for refraction, taking into consideration the readings of the thermometer and barometer, and apply the star's declination for the day (from the Nautical Almanac) for the latitude. The above instructions* apply only to stars observed near the meridian. The latitude can however, be obtained by similar observations of stars situated very far from the meridian, though this method would very seldom be resorted to. When the sun is the object observed, a further correction must be made on account of the change in declination during the time occupied by the observation, which is expressed by S being the change of declination in one minute of time, minus when decreasing. E the sum of the horary angles observed to the east, expressed in minutes of time, and considered as integers. W their sum to the west, and n the number of these observations. When a star is the object observed, and the time is noted by a chronometer, regulated to mean time, the value of A must be multiplied by 1.0054762. Also, if the clock does not keep its rate either of sidereal or mean time accurately, a further correction is imperative; and A must be multiplied by 1 + ·0002315 r, where r denotes the daily rate of the clock in seconds, minus when gaining, and plus when losing. * See "Corps Papers," vol. iii. page 328, where will also be found examples worked out in detail, of latitudes thus obtained on the survey of the North American Boundary. |