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To find the error of a chronometer on mean time at a place by EQUAL ALTITUDES of the sun.
When the sun's centre is on the meridian of any place, the apparent time is then either Oh or 24h. To obtain mean time at the same instant, we have only to apply the equation of time with its proper sign. We thus find mean time at the instant the sun is on the meridian; and if we can also ascertain what a chronometer showed at the same instant, it is manifest that the error of the chronometer on mean time at the place is known, since it will be the difference between the two times.
To find the time shown by the chronometer at apparent noon, we have recourse to the method of equal altitudes, which consists in noting the time shown by the chronometer when the heavenly body has the same altitude on both sides of the meridian: half the interval between the observations being added to what the chronometer showed at the first observation will be the time shown by the chronometer when the heavenly body is on the meridian, if the declination is supposed to be invariable in the interval between the observations.
For let t and t, be the times shown by the chronometer when the heavenly body is at x and y, at the same altitude on both sides of P Z the meridian; and suppose t, greater than t (that is, if the hour hand has arrived in the interval to 12h, we continue to count 13h, 14h, &c., instead of 1h, 2h, &c.). Now, if the rate of chronometer has been uniform
in the interval, the time elapsed is t, -t, and the heavenly body has described the angle z P X, or half y P x in the time To this half interval add the time t shown by the chronometer when the body was at I, we find that the
that the equation of time at noon is
4m 24s subtracted from apparent time; then mean time at the place when the sun is on the meridian is 24h
= 23h 55m 36. Now, if (t,+ t) is found to be 23h 10m 26s the difference 45m 10s is the error of the chronometer slow on mean time at the place.
But the sun's declination is not invariable during the interval t, — t, but increases or decreases by a small quantity, so that the angle zPx differs from half the interval by a few seconds.
The following rule enables us to find the number of seconds which must be applied to the half interval to obtain ZPX. This quantity of time is called the equation of equal altitudes.
To find the error of a chronometer on mean time by equal altitudes of the sun.
1. Find mean time nearly of apparent noon at the place by taking out of the Nautical Almanac the equation of time to the nearest minute, and applying it with its proper sign to Oh or 24h, according as the Nautical Almanac directs it to be added to or subtracted from apparent time, putting the day one back in the latter case.
2. To mean time nearly thus found apply the longitude in time, adding if west, and subtracting if east; the result will be a Greenwich date.
3. Correct the equation of time for this date.
4. From the P.M. time when the second altitude was taken (increased by 12 hours) subtract the A.M. time when the first altitude was taken; the remainder is elapsed time as shown by the chronometer: take half the elapsed time and subtract it from the above date (increased if necessary by 24 hours and the day put one back), the remainder is a second Greenwich date.
5. Take out the sun's declination for this date.
6. To find the equation of equal altitudes. Under heads (1) and (2) put down the following quantities.
Under (1) put A taken from annexed table.
(2) put B
(1) put log. cotangent latitude.
(2) put log. cotangent declination.
both (1) and (2) put proportional log. change of declination in 24 hours.
7. Add together logarithms under (1) and (2) and reject the tens in the index; look out the result as a proportional logarithm, and take out the seconds and tenths corresponding thereto.
8. Mark the quantities under (1) plus (+) if the declination is decreasing, and of the same name as the latitude; or, if increasing and of a different name. Otherwise mark the quantity minus (-).
9. Mark the quantity under (2) plus (+) if the declination is increasing, but minus (—) if decreasing.
10. Take the sum or difference of these quantities, according as they have the same or different signs; the result will be the correction or equation of equal altitudes required.
11. Add together A.M. time and half elapsed time, and to the same apply the correction just found with its proper
sign: the result will be the time shown by the chronometer when the sun's centre is on the meridian.
12. Find mean time at the same instant by applying the equation of time to Oh or 24h with the proper sign as directed in the Nautical Almanac.
13. Put down under each other the results determined in (11) and (12), and take the difference, which will be the error of the chronometer on mean time at the place.
14. To find error of the chronometer on Greenwich mean time. To mean time at the place as found in (12) apply the longitude in time, and thus get mean time at Greenwich, under which put the time shown by chronometer as found in (11); the difference will be the error of the chronometer on Greenwich mean time.
1.96649 1.99017 5 30 1.96541 1.99243 5 40 1.96426 1.99484 5 50 1-96305 1.99743 6 0 1.92866 2.07918 1.96176 2.00019 6 10 1.92604 2.08624 1.96040 2.00312 6 20 1.92333 2.09365 1.95897 2.00623 6 30 1.92054 2-10143 1-95747 2.00954 6 40 1-91767 2.10961 3 50 1.95589 2.01303 6 50 1.91473 2-11821 4 0 1.95424 2.01671 7 0 1.91170 2.12725 10 0 4 10 1.95252 2.02060 7 10 1.90859 2.13678 10 10 4 20 1.95073 2.02470 7 20 1.90539 2.14680 10 20