ever, is, on the whole, the more accurate, as allowance is made for the declinations of the sun and moon; and the number to be always subtracted is taken truly from the Table at the bottom of page 115, instead of applying the mean -22", as is done in method second. 4. To the time of upper transit, 5h 46m, and the moon's horizontal parallax, 58'.1, are obtained from Table LXIV., Part I., the factors 0.165a, and 0.9346. But by Table LX., a is 16 feet, and b is 9 feet, whence the height of the tide is 0.165 × 16+0 934 × 9= 11.046 feet. Also to lower transit 5" 19", and the moon's horizontal parallax 58′.5, the factors are 0.261a, and 0.8596, and the height will therefore be 11.907 feet. 5. Again, the height of the former tide, at 2 30 after high water, is from Part II., 11.046 × 0.67-7.40 feet, and that of the latter, 11.907 × 0.67-7.98 feet. TABLE LXIX. has been introduced for the purpose of computing the mean longitude of the moon's ascending node, which is necessary to determine the true places of the stars, contained in Table LIV., nearly comprehending all the fixed stars to the third magnitude inclusive. By means of this Table, and Table LXXIV., containing the sun's longitude, the arguments for entering the Tables, in order to determine the Nutation and Aberration, may be deduced without a reference to any other Tables, which may sometimes be inaccessible. The second column contains the mean longitude of the moon's ascending node on the first of January each year, on the meridian of Greenwich, when the sun's longitude is 281°, from 1820 till 1850. The remaining columns contain the retrograde motion of the node for months, days, and hours, which is always to be subtracted from the longitude of the given epoch. Ex. Required the longitude of the moon's ascending node on December 25, 1838, at 11 P. M., the time of upper culmination of Rigel? In this example the retrograde motion of the node for the mouths, days, and hours, being greater than the mean longitude at the beginning of the year, on the meridian of Greenwich, to which all our astronomical tables are adapted, 360° are to be added to it before the subtraction takes place. In the computations of the places of the stars, contained in the Nautical Almanac, no account has hitherto been taken of the 29th day of February in leap year, though, in cases where extreme accuracy is required, this ought not to be neglected. Should the longitude of the place for which the calculations are made, differ considerably from Greenwich, the time must be reduced to that of Greenwich, in the usual manner. Add 360.000 The degrees may be converted into signs, by dividing by 30°, and the value of the decimal will give the minutes near enough the truth, for the purpose here required. TABLE LXX. contains a correction of the moon's passage of the meridian of Greenwich, to reduce to it that of any other meridian. It must be added in west longitude, and subtracted in east, according to the title at the top. The arguments are the longitude in the left-hand column, and the daily difference of the times of passage, at the top. TABLE LXXI. contains the contraction of the semidiameters of the sun and moon, on account of refraction, according to the altitude of the object found in the left-hand column, and the inclination of the measured semidiameter at the top. TABLE LXXII. contains the effect of the solar nutation of a star in right ascension and declination, to be applied according to the signs. The second part of the solar nutation, in R. A., must be mul tiplied by the tangent of declination. TABLE LXXIII. consists of numbers for finding the length of a degree in latitude, longitude, and of the length of the pendulum at any latitude, to an ellipticity of 3, those on the equator being known. Length of an arc of one degree of latitude at the equator, is One degree of longitude there Length of equatorial pendulum 362755 feet. 365144 feet. 39.01326 inches. Whence, by multiplying any of these by the factor, for the latitude, from the Table, the result will be the length at the given latitude. Thus, at Edinburgh, in latitude 55° 57' 18" N. A degree of latitude = longitude = 362755 x 1.00685= 365144 x 0.56069= Length of pendulum = 39.01326 × 1.00370 365169 feet. 204733 feet. 39.15761 inches. = = 159.78 sec. TABLE LXXIV. gives the true longitude of the sun at apparent noon, on the meridian of Greenwich, for leap year, particularly 1828, and it may be adapted to other years, with sufficient precision, for several purposes, especially for finding the necessary arguments for computing the aberration of the fixed stars, &c., as exemplified in the explanation of Tables XLVI., XLVII., and XLVIII., by subtracting one-fourth of the difference between the given and preceding days from the longitude in the Table for the first after leap year, one-half for the second, and three-fourths for the third; or it will be near enough to subtract 15' for the first, 30' for the second, and 45′ for the third; and, in the months for January and February, the longitude is to be taken for the day following that given. TABLE LXXV. consists of the log. versed sine of the corresponding arc, or time diminished by the constant log. of 2, or 0.301030, and is therefore the logarithms of half the versed sines. But as 2 sine 2 P, is the versed sine of the arc P, it is evident that the Table is also the log. sine 2 P. If P be the angle at the pole, made by any meridian passing through a given object, with the meridian of the place of observation, the hour angle from noon will thus, from computation, become known. If Z, the angle at the zenith, be substituted for P, the azimuth will in like manner become known by the rules in page 99, illustrated by the examples in page 100. The quantities are taken from this Table, similarly to those in Table V.; and the proportional parts, for seconds of time, are very easily obtained from the differences to 100 by annexing two ciphers to the difference between the tabular and given logarithms and dividing this by the number in the column of differences; the quotient will be the seconds and decimal. As this 100 corresponds to 25', or the fourth part of 100',-to find the minutes, multiply the difference given in that column in the Table, by 4, annex two ciphers to the difference between the nearest logarithm found in the Table and the given logarithm, and, dividing the latter by the former, the result will be the minutes and decimal, which may be reduced to seconds if necessary; or, after annexing two ciphers, divide first by 4, and afterwards by the tabular difference, the same result will be obtained as before, and perhaps rather more simply. The following Tables, containing the solutions of Plane and Spherical Triangles, which, by Mathematicians conversant with Formula, will be always preferred to Rules, were drawn up by my friend, Mr Andrew Girvan, accountant, Edinburgh. In using these Formulæ, the signs of the Trigonometrical lines marked in the Table, page 12, must be carefully attended to. APPENDIX. Table for the Solution of Plane and Spherical Triangles. Let A, B, and C denote the three Angles of any Triangle, and a, b, and c the sides respectively opposite to these Angles. Given. I. Right-angled Plane Triangles, right angled at A. I. A side and an angle. a and B Required. Formulæ. b and c ba sin B, and ca cos B No cos (B-C) = 26 c a2 and 10 |