A sexagesimal is divided by subtracting the proportional log. of the divisor from the sum of the proportional logs. of the dividend and one minute. The proportional log. of a geometrical mean between two numbers is equal to half the sum of the proportional logs. of these numbers. That of the square root is found by taking half the sum of the proportional logs. of the given quantity and one minute; and the P. Log. of the square of a given quantity, by subtracting the P. Log. of one minute, from twice the P. Log. of the proposed quantity. If this quantity is greater than 13- its square will exceed 28 60 the limits of the table. In order, therefore, to extend this problem, subtract the proportional log. of 2.3.4, &c. minutes, from twice the proportional log. of the given quantity; the remainder will be the prop. log. of a half, third, fourth, &c. of the required quantity. Proportion is performed by subtracting the P. Log. of the first term from the sum of the P. Logs of the other two. If the first term is 32 or 3 hours, the sum of the P. Logs of the two last is that of the answer. If the first term is not 180', or 180°, it will be found convenient to reduce it, if possible, to that quantity, and one of the other terms must be increased or diminished in the same ratio. If either the second or third term is 180′, or 180°, the P, Log. of the fourth term is obtained by subtracting the P. Log. of the first term from that of the other. A sexagesimal is multiplied by the sine, tangent, or secant of an arch, by adding to its P. Log. the log. co-secant, co-tangent, or cosine of that arch; and a sexagesimal is divided by a sine, tangent, or secant, when the log. of the given sine, tangent, or secant, is added to its P. Logarithm. It may be remarked, that in the use of this table, one minute may also be esteemed either one degree or one second; by which means its use will be rendered more extensive. TABLES LII. LIII. LIV. LV. LVI. LVII. LVIII. LIX. LX. LXI. AND LXII. Of the Right Ascension and Declination of Fixed Stars. In the first of these tables are contained, the mean right ascension and declination of 188 fixed stars, adapted to the beginning of the year 1810, with their annual variations in right ascension and declination; by which, their mean place may be found with tolerable accuracy, for a few years preceding or following the epoch of the tables. The proper annual motions of a few of those stars are allowed for. The late celebrated astronomer, M. de la Lande, has given in the Connaissance des Tems, a table of the proper motions of 600 stars in right ascension and declination. The mean place of a star may be accurately ascertained for a future period, by applying its proper motion, and the equations from Tables LIII. and LIV. answerable to the given interval. Table LIII. contains part first of the precession in right ascension for complete years, common to all the stars. Part second of the precession in right ascension is found by multiplying the equation from Table LIII. answering to the given interval, and the right ascension of the star at the mean interval, by the natural tangent of its declination, and changing the sign, if the star is south of the equinoctial. The precession in declination is found by entering the same table with the right ascension of the star increased by 90°, and changing the sign, if the star is south. If the given interval is not found in the table, the sum of the equations answering to the years that make up the interval is to be taken. The precession in right ascension, answering to months and days, is found by multiplying the annual precession by the decimal from Table LV. answering to the given time; wherein the effect of the semi-annual equation of precession is allowed. The precession in declination, answering to months and days, is found by taking a part of the annual precession proportional to the time. The semi-annual solar equation of northern stars in declination, is contained in Table LVI. and the sign is to be changed, if the declination of the star is south.These two are taken from Dr. Maskelyne's tables. The nutation of a star in right ascension and declination is found by means of Tables LVII. LVII. and LIX. as follows: From the right ascension of the star, subtract the longitude of the Moon's node; the remainder will be the argument of Table LVIL. and their sum will be the argument of Table LVIIL. Multiply the sum or difference of the corresponding equations, according as they have the same or contrary signs, by the natural tangent of the star's declination, and change the sign of the product, if the declination is south; to which apply the equation from Table LIX, answering to the longitude of the Moon's node; and the sum or difference will be the nutation in right ascension. To each of the two first arguments, add three signs, and the sum or difference of the resulting equations, from Tables LVII. and LVIII. will be the nutation in declination. EXAMPLE. Required the nutation of a Aquila in right ascension and declination, January 1, 1815? R. A. a Aquilæ Tables LX. LX1. and LXII. for computing the aberration of a fixed star* in right ascension and declination, were constructed by M. de Lambre. The aberration may be found as follows: Subtract the Sun's longitude from the right ascension of the star the remainder will be the argument of Table 1.x. and their sum will be that of Table LXI. Now the sum or difference of the equations answering thereto, according as they have the same or contrary signs, being multiplied by the natural secant of the star's declination, will be its aberration in right ascension. Increase the argument of Tables LX. and LXI. each by three signs; take out the corresponding equations, and the product of their sum or difference by the natural sine of the star's declination, will be part first of the aberration in declination. The equation from Table LXII. answering to the sum of the Sun's longitude, and the declination of the star, is part second; and that from the same table answering to the Sun's longitude, diminished by the star's declination, is part third of the aberration in declination; which parts being connected according to their signs, will give the absolute aberration in declination. If the declination of the star is south, the sign of the final result is to be changed. EXAMPLE. Required the aberration a Aquila in right ascension and declination, January 1, 1815? In computing the place of a star, its right ascension and declination at the mean interval is to be used. These are found, by applying to the tabular place, the multiples of the annual variations answering to half the interval. Application of the preceding rules to the computation of the apparent right ascension and declination of a fixed star. *See Vol, I. page 21, EXAMPLE. Required the apparent right ascension and declination of Arcturus, 12th March, 1860. Mean R. A. Arcturus 1800,=211° 38' 3" Mean dec. 20° 13′ 44′′N. Variation in half interval, + 20 24 9 33 211 58 27 Dec.atm. int.20° 4 11 N. R. A. of Arcturus at beginning of 1800, Precession for days including the semi-annual equat. Mean right ascension, 12th March 1860, Apparent R. A. of Arcturus, 12th March 1860, Proper motion of Arcturus in decl. for 60 years, Precession in declination for days, Apparent declination of Arcturus, 12th March 1860, 19 54 24.8 TABLES LXIII. LXIV. LXV. LXVI. LXVII. AND LXVIII. Of the Longitude and Latitude of the Fixed Stars. The first of these tables contains the mean longitude and latitude of 122 fixed stars, adapted to the beginning of the year 1810. The longitude and latitude of 34 of these stars were computed from a table containing their right ascension, communicated by Dr. Maskelyne; and the places of the other stars were inferred from the catalogues of Messrs. Bradley and Mayer. The stars in this table are mostly zodiacal, being intended principally for ascertaining the longitude, from occultations of fixed stars by the Moon. The apparent longitude and latitude of a star may be found at any given time from its mean place, by means of the five following tables. The first of them, or Table LXIV. contains the precession in longitude for completer ears, and Table LXV. contains the precession for months and and days. The secular equation in longitude is found by multiplying the equation from Table LXVI. answering to the longitude of the star, by the natural tangent of the star's latitude, and to the product, applying the constant quantity -15".4. The equation from the same table, answering to the longitude of the star, increased by 3 signs, will be the secular equation in latitude. The equation answering to any number of years is found by taking the proportional part of the secular equation. Table LXV I. contains the equation of the equinoxes in longitude, common fixed stars and planets; and in Table LXV:11. is the aberration of points of the ecliptic in longitude; from which the aberration of a star in longitude and latitude may be found thus: From the Sun's longitude subtract that of the star; find the corresponding equation, which multiplied by the natural secant of the star's latitude, will be the aberration in longitude. Increase the former argument by 3 signs, and the equation answering thereto, being multiplied by the natural sine of the star's latitude, will be the aberration of the star in latitude. In computing the apparent place of a star for a proposed time, its mean longitude, as settled at some certain epoch, must be correct-, ed by the precession of the equinoxes in longitude, answering to the interval, from Tables LXIV. and LXV; the proportional part of the secular variation, Table LXVI.; the equation of the equinoxes in longitude, Table LXVII; and the aberration, Table LXVIII. The apparent latitude is found by applying the proportional part of the secular equation, and the aberration. The proper motion of the star in longitude and altitude should also be applied. EXAMPLE. What is the apparent longitude and latitude of a Andromeda, 1st June 1814? Mean longitude of a Andromedæ, beginning of 1800, α The maxima of the equations of the above tables for correcting the mean place of a star, are agreeable to the determination of Drs, Brad ley |