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To find the longitude of a place, from the observed mean times of beginning and end of an eclipse of the sun, at that place.

Taking, for the place, an assumed longitude as nearly correct as the knowledge of its situation permits, reduce the observed time of beginning to Greenwich time, and for this time, find the equation of time from the Nautical Almanac. Apply the equation of time, according to the direction at the head of its column, to the observed time of beginning, and the result will be the apparent time of beginning. The interval between this time and noon, marked negative when the time is before noon, will, when converted into degrees, be the hour angle H.

Let T the Greenwich time of beginning, taken to the nearest whole minute; and for the time T', find, as in the last problem, p, q, p', q, t, u, v, and h, omitting u' and v', which are not required. Then, using p' and q instead of (p' — u′) and (q′ u′) and (g′ — v′), find the corrected Greenwich time of beginning, as in the last Problem. The difference between this corrected Greenwich time and the observed time of beginning, will be the longitude of the place in time, as deduced from the observed beginning; to the west when the observed time is the earlier of the two, but to the east when it is the later. If the longitude thus obtained differs several minutes from the assumed longitude, the calculation should be repeated, taking the longitude obtained for the assumed longitude.

In a similar manner find the longitude from the observed time of the end. The half sum of the two results will the longitude of the place as given by the observations of both beginning and end.

Note 1. When a table of the values of p, q, &c., has not been previously calculated, the values of p, q, and 1, may be computed from the formulæ for the time T' at beginning and the time T' at the end. The value of p, at the time T' for beginning, subtracted from its value for the time T'at the end, will be the change of value during the interval between these two times; from this, the value of p', the hourly change of value of p, may be easily obtained, with sufficient accuracy, by proportion. In the same way, the value of q' may be found from the two computed values of q.

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2. When the eclipse has been observed at places whose longitudes are accurately known, corrections of the computed longitude, due to errors in the tables, may be obtained by the method in Art. 82 of the Appendix.


The observed beginning of the eclipse of May 15th, 1836, at Haverford school, latitude 40° 1' 12′′ N., and assumed longitude 5 h. 1 m. 25 sec.W. was at 7 h. 3 m. 24.5 sec. A. M., mean time; and the end, at 9 h. 31 m. 47 sec. Required the longitude.


Observed time of beginning
Assumed diff. of long., add

Greenwich time of beginning

a = .00096

Observed time of beginning
Equation of time, add


log. p cos. ¿'


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d. h. m.

At T'

15 0 5

From the table of values in the example in the last problem, we find, p=-1.09295;q=0.10663; p=0.4807; q'=+0.1738; 7.56482 log.p sin p'

9.8059 log. p sin '

. 9.80595

log.cos 9.97582



log. cos 9.46184

b = .00097



2 c2

log. A. 7.1746



log. sin 9.98098 n

9.86571 n

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N=70° 7′ 19′′ log. cot 9.55818
.35893 log. 9.55501 n

P - u



9.11319 n

d. h. m. sec. 14 19 3 24.5 5 1 25

15 0 4 49.5


d. h. m. sec.

14 19 3 24.5

3 56.05

14 19 7 20.55

4 52 39.45

73° 9′52′′.

log. p cos o'



f = .60501

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log. sin 9.97332 dv-q=29649 log. 9.47201 h . . Ar. Co. 66 0.24958 F-60° 18' 27" log.cos 9.69491

N+F=130° 25′ 46′′ log.cos 9.81192 n p-u ·log. 9.75042


Ar. Co." 0.31813


9.88047 n



t =



-ť + t 15

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Observed time of end,
Assumed longitude, add

Greenwich time of end

Observed time of end,
Equation of time, add


log. p cos p



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h. m.


0 4 14.2

14 19 3 24.5

5 0 49.7 W longitude, from observed beginning.

d. h. m. sec.

14 21 31 47

5 1 25

15 2 33 12

At T'

p=0.09312;q=0.53452; p = 0.4810; q

log. p sin '

log. p sin o'

= .00270




log. A. 7.1752


d. h. m.

15 2 33

corrected Greenwich time of beginning.

observed time


log. sin 9.76996 n

9.65469 n

- 0.74668

log. G 9.7923 log. B 7.6390



9.88473 log. p cos p'


Ar. C.


D' .

log. 9.55501 n 0.31813

9.87314 n

d. h. m. sec.

14 21 31 47

ƒ= .60492

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0.1731; 7 0.56500


log. cos 9.97575




v = .40344


log. cos 9.90757

log. G 9.79230 log. sin 9.51194


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log. sin 9.97355 dv-q=.06492 log. 8.81238


Ar. Co." 0.25077

log. 9.73612 F=83° 45′ 10′′ log cos. 9.03670



N = 70° 12′ 27′′ log. cot 9.55615


u = .54465

d = .19600

log. 9.23830


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10° 32′ 43′′ log. cos 9.98775

log. 9.74923

A. C." 0.31785


d. h. m.

Ttt 15 2 33

half sum

14 21 31 47




1 21.1 W 5 0 49.7 W

= 5 1 5.4 W


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p U log. 9.73612
A. C.



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Scholium. The longitude thus obtained is subject, however, to the error which results from errors in the tables. But the present eclipse being visible and observed, at many of the European Observatories, as well as in this country, the longitudes of which had been previously ascertained with considerable accuracy, the means have been afforded for correcting this error, by the method in the Appendix (82). C. Rumker, Director of the Hamburg Observatory, computed the principal observations made both in Europe and this country, and thence obtained equations for correcting the errors of the tables. From these, Sears C. Walker, of Philadelphia, has obtained 2′′.934, and 7".198.* & = With these values and those of a and b, which are easily found from their expressions (App. 82 y), we obtain (App. 82 z), -15.22 sec., and -4.64 sec., for the corrections to be added to the longitudes found above, from the observed beginning and end respectively. Since the longitudes are west, they are, in accordance with the formula, to be regarded as negative. Hence the corrected longitude deduced from the observed beginning is 5 h. 1 m. 4.92 sec. W., and that from the end, 4 h. 1 m. 25.74 sec. W. ;

* Transactions of the American Philosophical Society, vol. VI., new series.

the mean of which, 5h. 1m. 15.33sec. W., is the longitude of Haverford School, as given by the observations.*

The observations of the eclipse made in this country, combined with those made in Europe, afforded favourable means for determining the moon's parallax. The constant of her equatorial horizontal parallax, deduced by S. C. Walker, is 57′ 2′′; which, agreeing very nearly with a late determination of its value by Henderson, from an extensive series of meridian observations made at Greenwich, Cambridge, and the Cape of Good Hope, is probably to be regarded as more accurate than 57' 1", given in the former part of the work (95).

To calculate an Occultation of a fixed star by the moon, for a given place.

1. Let A moon's right ascension, A'star's right ascension, D moon's declination, D' star's declination, A" moon's hourly variation in right ascension, D' moon's hourly variation in declination, ♫ — moon's equatorial horizontal parallax, H' star's hour angle for Greenwich, and H star's hour angle for the given place.




2. Let T = the mean time of conjunction of the moon and star in right ascension, taken to the nearest whole hour; and for the time T, find the quantities p, q, p', and q', from the following formula.


D - D'


A') cos D


A" cos D




The quantities p' and q', which are the hourly variations of p and q, may be regarded as constant. The values of p and q for a time T', may be found by adding to, or subtracting from, their values at the time T, the quantities (T∞ T′). p', and (T∞ T′). 9′, respectively, according as T' is later or earlier than T.


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3. To the logarithm 9.4192, and from it, add and subtract log. sin D', and the sum and remainder will be respectively two logarithms of D and E,

4. To the sidereal time at mean noon at Greenwich, on the day of the occultation, taken from the Nautical Almanac, add the sidereal time cor

* From the eclipse of September, 1838, Prof. Kendall, who computed the longitudes of various places at which the eclipse was observed, obtained for that of Haverford School, 5h. 1m. 15.0sec. W.-Am. Philos. Trans. vol. VII.

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