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common logarithm of this, subtract the constant log. 2. 101510*, and the remainder will be the log. of the meridional parts answering to that latitude.

Example 1.

Required the meridional parts corresponding to latitude 50:48? ? Given lat. 50:48; complement 39:12÷2= =

=

complement; hence,

=

19:36, the half

Half comp. 19:36′ log. co-tangent less radius = .448448, the log. of which is

Constant log.

5.651712 2. 101510

Meridional parts corresponding to given lat. 3549.78=log.=3.550202

Example 2.

Required the meridional parts corresponding to latitude 89:30?? Given lat. 89:30; comp. 0:30:20:15, the half comple

ment; hence,

Half comp. 0:15 log. co-tangent less radius = 2.360180, the log.

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Meridional parts corresponding to given lat. 18682. 49=log.=4.271435

TABLE XLIV.

The Mean Right Ascensions and Declinations of the principal fixed Stars.

This Table contains the mean right ascensions and declinations of the principal fixed stars adapted to the beginning of the year 1824.-The stars are arranged in the Table according to the order of right ascension in which they respectively come to the meridian; the annual variation, in right ascension and declination, is given in seconds and decimal parts of a second; that of the former being expressed in time, and that of the latter motion.

The stars marked †, have been taken from the Nautical Almanac for the year 1824.-The stars that have asterisks prefixed to them are those from which the moon's distance is computed in the Nautical Almanac for the purpose of finding the longitude at sea.

The measure of the arc of 1 minute (page 54,) is .0002908882; which being multiplied by 10000000000, (the radius of the Tables) produces 290.8882000000; and, this being multiplied by the modulus of the common logarithms, viz., 43429448190, gives 126.331140109823580; the common log, of which is 2.101510, as above.

The places of the stars, as given in this Table, may be reduced to any future period by multiplying the annual variation by the number of years and parts of a year elapsed between the beginning of 1824, and such future period the product of right ascension is to be added to the right ascensions of all the stars, except 8 and 8, in Ursa Minor, from whose right ascensions it is to be subtracted: but the product of declination is to be applied, according to the sign prefixed to the annual variation in the Table, to the declinations of all the stars without any exception;-thus,

To find the right ascension and the declination of a Arietis, Jan. 1st, 1884.

R. A. of a Arietis, per Tab. 15716, and its dec.

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Rt. asc. of a Arietis, as req. 1:57:49'.5, and its declination 22:40:27" N.

Should the places of the stars be required for any period antecedent to 1824, it is evident that the products of right ascension and declination must be applied in a contrary manner.

The eighth column of this Table contains the true spherical distance and the approximate bearing between the stars therein contained and those preceding, or abreast of them on the same horizontal line; and the ninth, or last column of the page, the annual variation of that distance expressed in seconds and decimal parts of a second.-By means of the last column, the tabular distance may be reduced very readily to any future period, by multiplying the years and parts of a year between any such period and the epoch of the Table, by the annual variation of distance; the product being applied by addition or subtraction to the tabular distance, according as the sign may be affirmative or negative, the sum or difference will be the distance reduced to that period.

Example.

Required the distance between a Arietis and Aldebaran, Jan. 1st, 1844 ?

Tabular dist. between the

Annual var. of distance.

α

two given stars = .

Number of years after 1824 =

Product.

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20

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True spherical distance between the two given stars, as

required..

35:32:6".60.

Remark. The true spherical distance between any two stars, whose right ascensions and declinations are known, may be computed by the following rule; viz.,

To twice the log. sine of half the difference of right ascension, in degrees add the log. sines of the polar distances of the objects; from half the sum of these three logs. subtract the log. sine of half the difference of the polar distances, and the remainder will be the log. tangent of an arch; the log. sine of which being subtracted from the half sum of the three logs., will leave the log. sine of half the true distance between the two given stars.

Example.

Let it be required to compute the true spherical distance between ɑ Arietis and Aldebaran, January 1, 1844.

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Half difference of R. A. in de-318:34:27" Twice the}19.0063060

grees.

N. polar dist. of a Arietis = {67.16.39
N, polar dist. of Aldebaran = {73.48.32

Log. sine

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Diff. of Polar dists. 6:31:53 Half=19.4768212

Half diff. of ditto 3:15756 Log S. 8.7556177/

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Arch 79:14:27". 5826 log. tang. .10.7212035 Log. S. 9.9922976.3

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Now, by comparing this computed distance with that directly deduced from the Table, as in the preceding example, it will be seen that the difference amounts to very little more than the fifth part of a second in twenty. years; which evidently demonstrates that the tabular distances may be reduced to any subsequent period, for a considerable series of years, with all the accuracy that may be necessary for the common purposes of navigation.

Note. The tabular distances will be found particularly useful in determining the latitude, at sea, by the altitudes of two stars, as will be shown hereafter.

TABLE XLV.

Acceleration of the Fixed Stars; or to reduce Sidereal to Mean Solar Time.

Observation has shown that the interval between any two consecutive transits of a fixed star over the same meridian is only 23 564.09, whilst that of the sun is 24 hours :-the former is called a sidereal day, and the latter a solar day; the difference between those intervals is 355.91, and which difference is called the acceleration of the fixed stars.

This acceleration is occasioned by the earth's annual motion round its orbit: and since that motion is from west to east at the mean rate of 59.8".3 of a degree each day; if, therefore, the sun and a fixed star be observed on any day to pass the meridian of a given place at the same instant, it will be found the next day when the star returns to the same meridian, that the sun will be nearly a degree short of it; that is, the star will have gained 3 56.55 sidereal time, on the sun, or 3" 55′ .91 in mean solar time; and which amounts to one sidereal day in the course of a year :-for 355.91 x 36554848: 23456" 4::-hence in 365 days as measured by the transits of the sun over the same meridian, there are 366 days as measured by those of a fixed star.

=

Now, because of the earth's equable or uniform motion on its axis, any given meridian will revolve from any particular star to the same star again in every diurnal revolution of the earth, without the least perceptible difference of time shewn by a watch, or clock, that goes well :-and this presents us with an easy and infallible method of ascertaining the error and the rate of a watch or clock :-to do which we have only to observe the instant of the disappearance of any bright star, during several successive nights, behind some fixed object, as a chimney or corner of a house at a

little distance, the position of the eye being fixed at some particular spot, such as at a small hole in a window-shutter nearly in the plane of the meridian; then if the observed times of disappearance correspond with the acceleration contained in the second column of the first compartment of the present Table, it will be an undoubted proof that the watch is well regulated :—hence, if the watch be exactly true, the disappearance of the same star will be 356. earlier every night; that is, it will disappear 3*56! sooner the first night; 752: sooner the second night; 1148: sooner the third night, and so on, as in the Table.-Should the watch, or clock deviate from those times, it must be corrected accordingly; and since the disappearance of a star is instantaneous, we may thus determine the rate of a watch to at least half a second.

The first compartment of this Table consists of two columns; the first of which contains the sidereal days, or the interval between two successive transits of a fixed star over the same meridian, and the second the acceleration of the stars expressed in mean solar time; which is extended to 30 days, so as to afford ample opportunities for the due regulation of clocks or watches. The five following compartments consist of two columns each, and are particularly adapted to the reduction of sidereal time into mean solar time-the correction expressed in the column marked acceleration, &c. being subtracted from its corresponding sidereal time, will reduce it to mean solar time; as thus.

Required the mean solar time corresponding to 14:40 55 sidereal time?

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Remark. This Table was computed in the following manner; viz.,

Since the earth performs its revolution round its orbit, that is, round the sun, in a solar year; therefore as 365 548 48: 360::: 1:59:8".3; which, therefore, is the earth's daily advance in its orbit: but while the earth is going through this daily portion of its orbit, it turns once round on its axis, from west to east, and thereby describes an are of 360:59:8".3 in a mean solar day, and an arc of 360: in a sidereal day.

Hence, as 360°:59:8".3: 24::360: 23:56" 4'.09, the length of a sidereal day in mean solar time; and which, therefore, evidently anticipates 355.91 upon the solar day as before-mentioned. Now,

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