of collimation at top; opposite the former, and under the latter, will be found the corresponding correction. Thus, if the observed angle be 80 degrees, and the inclination of the line of collimation 30 minutes, the corresponding error will be 13 seconds. The error or correction taken from this Table is always to be applied by subtraction to the observed angle. The corrections in this Table were computed by the following Rule. To the log. sine of half the observed angle, add the log. cosine of the inclination of the line of collimation; and the sum, rejecting 10 in the index, will be the log. sine of an arch. Now, the difference between twice this arch and the observed angle, will be the error of the line of collimation. Example. Let the observed angle be 80 degrees, and the inclination of the line of collimation 1:30! ; required the corresponding correction? Difference 0: 1:58", which, therefore, is the required error, TABLE XXIV. Logarithmic Difference. This Table contains the logarithmic difference, adapted to every tenth minute of the moon's apparent altitude from the horizon to the zenith, and to each minute of horizontal parallax. The proportional part for the excess of the given above the next less tabular altitude, is contained in the right-hand compartment of each page, and that answering to the seconds of parallax is given in the intermediate part of the Table. As the size of the paper would not admit of the complete insertion of the logarithmic difference, except in the first vertical column of each page, under or over 53:, therefore in the eight following columns it is only the four last figures of the logarithmic difference that are given : hence, in taking out the numbers from these columns, they are always to be prefixed by the characteristic, and the two leading figures in the first column. The logarithmic difference is to be taken out in the following manner. Enter the Table with the moon's apparent altitude in the left-hand column of the page, or the altitude next less if there be any odd minutes, opposite to which, and under the minutes of the moon's horizontal parallax, at top, will be found a number, which call the approximate logarithmic difference. Enter the compartment of the “ Proportional parts to seconds of parallax," abreast of the approximate logarithmic difference, with the tenths of seconds of the moon's horizontal parallax in the vertical column, and the units at the top, and take out the corresponding correction. Enter the right-hand compartment of the page, * abreast of where the approximate logarithmic difference was found, or nearly so, with the odd minutes of altitude, and take out the corresponding correction, which place under the former. Enter Table XXV or XXVI., with the sun's, star's, or planet's apparent altitude, and take out the corresponding correction, which also place under the former. Now, the sum of these three corrections being taken from the approximate logarithmic difference, will leave the correct logarithmic difference, Example 1. Let the moon's apparent altitude be 19:25:, her horizontal parallax 60'38", and the sun's apparent altitude 33 degrees; required the logarithmic difference? Log. difference to app. alt. 19:20", and hor. par. 60! is 9.997669 49 Cor, from Tab. XXV. ans. to sun's apparent alt. is 10 11 sum In taking out the correction corresponding to the odd minutes of altitude in this comipartment, attention is to be paid to the moon's horizontal parallax : thus, if the parallax be between 53 and 56', the correction is to be taken out of the first column, or that adjoining the minutes of altitude ; if it be between 56' and 59', the correction is to be taken out of the second, or middle column; and if it be between 59' and 62'; the correction is to be taken out of the third, or last column. Example 2. Let the moon's apparent altitude be 63:37', her horizontal parallax 58:43", the apparent altitude of a planet 35:10:, and its horizontal parallax 23"; required the logarithmic difference? Log. difference to appar. alt. 63:30!, and hor. par. 58', is 9.993622 Propor. part to 43! of parallax is 83 Propor. part to 75 of altitude is 7 sum =-118 Cor. from Tab. XXVI. ans, to planet’s appar. alt. 28 Logarithmic difference, as required 9. 993504 Remark.-The logarithmic difference was computed by the following Rule. To the logarithmic secant of the moon's apparent altitude, add the logarithmic cosine of her true altitude, and the constant log. .000120;* the sum of these three logs., abating 10 in the index, will be the logarithmic difference. Example. Let the moon's apparent altitude be 19:20', and her horizontal parallax 60 minutes; required the logarithmic difference? * The difference between the log. cosines of the true and apparent altitude of a star betwixt 30 and 90 degrees. TABLE XXV. Correction of the Logarithmic Difference. This Table is divided into two parts: the first, or left-hand part, contains the correction of the logarithmic difference when the moon's distance from the sun is observed ; and the second, or right-hand part, the correction of that log. when the moon's distance from a star is observed. Thus, if the sun's apparent altitude be 35 degrees, the corresponding correction will be 11 ; if a star's apparent altitude be 20 degrees, the corresponding correction will be l; and so on. These corrections are always to be applied by subtraction to the logarithmic difference deduced from the preceding Table. The corrections contained in this Table were obtained in the following viz, To the log. secant of the apparent altitude, add the log. cosine of the true altitude; and the sum, rejecting 10 from the index, will be a log.; which being subtracted from the constant log. .000120,* will leave the tabular correction. manner, Example 1. Let the sun's apparent altitude be 35 degrees ; required the tabular correction? Example 2. Let the apparent altitude of a star be 10 degrees ; required the tabular correction See Note, page 50. Star's apparent altitude 100! 0! Log. secant = 10.006649 Correction of the Logarithmic Difference when the Moon's Distance from a Planet is observed. The arguments of this Table are, the apparent altitude of a planet in the left or right-hand marginal column, and its horizontal parallax at top; in the angle of meeting stands the corresponding correction, which is to be applied by subtraction to the logarithmic difference deduced from Table XXIV., when the moon's distance from a planet is observed. Hence, if the apparent altitude of a planet be 20 degrees, and its horizontal parallax 21 seconds, the corresponding correction will be 16 subtractive, and so on. This Table was computed by the rule in page 51, under which the correction corresponding to the sun's apparent altitude in Table XXV. was obtained, as thus : Let the apparent altitude of a planet be 23 degrees, and its horizontal parallax 21 seconds ; required the correction of the logarithmic difference? |