answering to that course (towards the right hand or the left, according as the first tabular difference of latitude which meets the eye therein is greater or less than the given departure), and find a difference of latitude that either agrees with, or comes nearest to, the given departure; then the distance corresponding to this, at the top or bottom of the column, will be the difference of longitude. Example. The middle latitude between two places is 20: north, and the meridian distance or departure 140 miles; required the difference of longitude ? Solution. The middle latitude, 20 degrees, taken as a course, and the departure 140, as difference of latitude, will be found to correspond in the compartment under or over 149: hence the difference of longitude is 149 miles, as required. PROBLEM VI. Given the middle Latitude, the Difference of Latitude, and the Difference of Longitude between two Places, to find the Course and Distance. RULE. Enter the Table with the difference of longitude, esteemed as distance, at the top or bottom of the page, and the middle latitude, taken as a course, in the left or right-hand column; answering to which, in the difference of latitude column, will be found the departure. Now, with this departure and the given difference of latitude, the course and distance are to be found by Problem II. Example. The middle latitude is 26 degrees north, the difference of latitude 200 miles north, and the difference of longitude 208 miles east; required the course and distance? Solution. In the compartment under or over 208 miles (the given longitude), and opposite to 26 degrees (the middle latitude taken as a course), stands 186.9 in the difference of latitude column, which, therefore, is the departure. Now, the tabular numbers answering nearest to the given difference of latitude and the departure, thus found, are 200. 4 and 186.9 respectively; these are found in the compartment under or over 274, and opposite to 43 degrees: hence the course is N. 43: E., and the distance 274 Remark. The numbers in the general Traverse Table were computed agreeably to the following rule; viz., As radius is to the distance, so is the co-sine of the course to the difference of latitude; and so is the sine of the course to the departure. Example. Given the course 35 degrees, and the distance 147 miles; to compute the difference of latitude and the departure. This Table contains the meridional parts answering to each degree and minute of latitude from the equator to the poles; the arguments of which are, the degrees at the top, and the minutes in the left or right hand marginal columns; under the former, and opposite to the latter, in any given latitude, will be found the meridional parts corresponding thereto, and conversely. Thus, if the latitude be 50:48, the corresponding meridional parts will be 3549.8 miles. Remark. The Table of meridional parts may be computed by the following rule; viz., Find the logarithmic co-tangent less radius of half the complement of any latitude, and let it be esteemed as an integral number; now, from the 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 complement; hence, Half comp. 0:15: log. co-tangent less radius = 2.360180, the log. 6.372945 2. 101510 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, 1834. R. A. of a Arietis, per Tab. 1:5716, and its dec. Ann. var.+17".40. after 1824 = 10 Prod.+174".0 + 2:54% 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. α Required the distance between a Arietis and Aldebaran, Jan. 1st, 1844 ? 35:32:7" 0".40 True spherical distance between the two given stars, as 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 degr ees 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 « Arietis and Aldebaran, January 1, 1844. = R. A. of a Arietis red. to 1844 158 23:, and its dec. =22:43:21" N. R. A. of Aldebaran red. to 1844 4. 26. 58. 6, and its dec. 16. 11. 28 N. = 37:8:54"+2=18:34:27" Log. sine 19.0063060 Lor.} sine {L} 9.9824236 sine Sum.. 38. 9536425 Diff. of Polar dists. 6:31:53 Half=19.4768212 Half diff. of ditto 3:15756% Log S. 8.7556177/ Arch 79:14:27". 5826 log. tang. . 10.7212035 Log. S. 9.9922976.3 |