EQUATION OF EQUAL ALTITUDES FOR FINDING THE ERROR OF THE CHRONOMETER In the case of a star the easterly and westerly hour-angles corresponding to the equal altitudes will be the same, and consequently half the sum of the times by chronometer when the observations were made is the exact time of the meridian passage of the star; but for the sun and planets the easterly and westerly hour-angles are unequal, in consequence of the change of declination in the interval. Half the sum of the times by chronometer when the observations were made is not, therefore, the correct time of meridian passage. The correction to be applied to the half-sum of the chronometer times to give the exact time of the meridian passage is called the Equation of Equal Altitudes. The Observation by the Sun.—On shore, at a place whose longitude is accurately known, and whose latitude is approximately known, observe with an artificial horizon the same altitude both in the morning and in the afternoon, as near the prime vertical as convenient after the altitude is more than 10°, noting the times by a chronometer. In low latitudes, however, the method of equal altitudes will often give very accurate results, even when the observations are quite near the meridian. In general, a sufficiently accurate result may be obtained if the observations are taken when the sun's change of altitude is not less than 10" in half a second of time, or when the change in the altitude taken with the artificial horizon is not less than 20" in half a second of time. It is most convenient, as well as conducive to accuracy, to take the observation in the following manner. In the morning bring the lower limb of the sun, reflected from the sextant mirrors, and the upper limb of that reflected from the mercury, into approximate contact; move the o (zero) of the vernier forward (say from 10' to 20'), and set it on a division of the limb; the images will be over-lapped and will be separating; wait for the instant of contact, note it by chronometer, and immediately set the vernier on the next division of the limb, that is io' in advance; notice the instant of contact again, and proceed in the same manner for as many observations as are thought necessary. If the sun rises too rapidly let the intervals on the limb be 20'. Find (roughly) the time when the sun will be at the same altitude in the afternoon, and just before that time set the vernier on the last altitude noted in the morning (of course using the same sextant); the images of the sun will be separated, but will be approaching; wait for the instant of contact, note it by chronometer, set the vernier back to the next division of the limb (10' or 20' as the case may be), note the contact again, and so proceed until all the a.m. altitudes have been again noted as p.m. altitudes. RULE.--For the Greenwich Date.-To the date at place (noon) ) oh. om. os. apply the longitude in time, additive if W., subtractive if E.; the result will be the Greenwich apparent date. From the Nautical Almanac, p. I., take out the sun's declination and the equation of time, and correct them for the Greenwich apparent date. For the Interval and the Middle Time by Chronometer.—Find the mean of the first times of observation shown by chronometer, and also the mean of the second times by chronometer ; take the first mean from the second mean for the interval between the observations. Then take the half of the interval, and add it to the mean of the first times by chronometer; the result will be the middle time by chronometer. For the Change of Declination in Method 1.-From Nautical Almanac write down the sun's declination for the day preceding and the day following the date of observation; take their difference and reduce the result to seconds ("), for the change of declination in two days. NOTE.-In March and September when the declination has changed its name, the sum of the declinations may be their difference. For the Change of Declination in Method II.-Multiply the “ Var. in I hour" by the half-elapsed time. Method I. RULE.-1. For the first part of the equation of equal altitudes, add together, Log. A of the interval (Table for Computing the Equation of Equal Altitudes in Norie's Tables). Log. of the seconds in the change of declination in two days, and The sum will be the log. of the first part, and take out the number corresponding thereto. For the second part of the equation of equal altitudes, add together-Log. B of the interval (Table for Computing the Equation of Equal Altitudes in Norie's Tables). Log. of the seconds in the change of declination in two days, and The sum will be the log. of the second part and take out the number corresponding thereto. 2. Method II. RULE.-1. For the first part of the equation of equal altitudes, add together Log. of the seconds in the change of declination in half-elapsed time. The sum, rejecting 20 from the index, will be the logarithm of the first part, and take out the number corresponding thereto. For the second part of the equation of equal altitudes, add together Log. of the seconds in the change of declination in half-elapsed time. Log. co-tangent of half-elapsed time. Log. tangent of the declination. 2. The sum, rejecting 20 from the index, will be the logarithm of the second part, and take out the number corresponding thereto. Formula e = c X co-sec. h x tan. I-CX cot. h x tan. d where e is the equation of equal altitudes, c is the change of declination in half-elapsed time, h is the half-elapsed time, I is the latitude, and d the declination. For both Methods.—Mark the first part + when the polar distance is increasing, and — when it is decreasing. Mark the second part + when the declination is increasing, and — when it is decreasing When the parts have the same sign their sum is the equation of equal altiludes, of the same sign as the parts. When the parts have different signs their difference is the equation of equal altitudes, of the same sign as the greater. Note to Method II.—The equation of equal altitudes is in seconds of arc, it must therefore be divided by 15 to give seconds of time. The application of the equation of equal altitudes to the middle time by chronometer gives the corrected middle time by chronometer when the sun was on the meridian of the place of observation. For the Error of the Chronometer on Mean Time at Place.To the date at place d. oh. om. os. apply the equation of time according to precept, Nautical Almanac, p. I., the result will be mean time at place, the difference between which and the corrected middle time by chronometer will be the error of Chronometer on mean time at place of observation. For the Error of the Chronometer on Greenwich Mean Time.—To the mean time at place apply the longitude in time, additive if W., but subtractive if E.; the result will be mean time at Greenwich, the difference between which and the corrected middle time by chronometer will be the error of the chronometer on Greenwich mean time on the given date. Example.—March 2nd : lat. 32° 2' N., long. 81° 3' W.; the following times by chronometer were noted when the sun had equal altitudes. Required the equation of equal altitudes, also the error of the chronometer on mean time at place, and on mean time at Greenwich. Example.—June 16th: lat. 10° 26' N., long. 45° 1' E. : the following times by chronometer were noted when the sun had equal altitudes S . ist part 2nd part 6.03 log. 0.7802 15)1.83 Diff. because the signs are unlike. EQUAL ALTITUDES OF A FIXED STAR. The Observation.-Set the sextant and wait for the coincidences of the two images of the star, as in the case of the sun's limb, noting the times by chronometer. The Computation.—Take the mean of the times before the meridian passage, also the mean of the times after the meridian passage. The mean of the two sets of times is the chronometer time of the star's transit. This time, if the chronometer is right, will agree with the true mean time of the star's transit, which is found as follows To the star's right ascension apply the longitude (in time) of the place, adding in W., subtracting in E. ; the result is the Greenwich sidereal time of star's transit ; from this subtract the sidereal time at the previous mean noon at Greenwich (Nautical Almanac, p. II.); the remainder is the sidercal interval since mean noon. From the Table for Reducing Sidereal to Mean Time, “ Retardation," with the argument sidereal interval take out the correction, which subtract from the sidereal interval; the remainder is the Greenwich mean time of star's transit. The time by chronometer will be more or less than this, according as the chronometer is fast or slow. |