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parallax ; with proportional parts adapted to the intermediate minutes of altitude, and to the seconds of horizontal parallax.
This Table was calculated in the following manner :
To the moon's apparent altitude apply the correction from Table XVIII., and the sum will be her true altitude; from the log. cosine of which (the index being augmented by 10) subtract the log. cosine of her apparent altitude, and the remainder will be a log., which, being diminished by the constant log. .300910,* will give the logarithmic cosine of the auxiliary angle.
Let the moon's apparent altitude be 4 degrees, and her horizontal parallax 55 minutes; required the corresponding auxiliary angle?
Moon's apparent altitude . 4. 0! 0 Log. cosine
Auxiliary angle, as required 60: 1:21"= Log. cosine
. 9. 698676
The correction of the auxiliary angle for the sun's or star's apparent altitude, given at the bottom of each page of the Table, was computed by the following rule—viz.
From the log. cosine of the sun's or star's true altitude subtract the log. cosine, of the apparent altitude, and find the difference between the remainder and the constant log. .000120.+ Now this difference, being subtracted from the log. cosinè of 60' degrees, will leave the log. cosine of an arch; the difference between which and 60 degrees will be the correction of the auxiliary angle depending on the apparent altitude of the sun or star.
Let the sun's or star's apparent altitude be 3 degrees; required the correction of the auxiliary angle?
• This is the log. secant, less radius, of 60 degrees diminished by .000120, the difference between the log. cosines of a star's true and apparent altitude betwixt 30 and 90 degrees.
+ This is the difference between the log, cosines of a star's true and apparent altitude, between 30 and 90 degrees.
0: 0: 8"; which, therefore, is the required correction of the auxiliary angle.
In this Table the auxiliary angle is given to every tenth minute of the moon's apparent altitude (as has been before observed) 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 column of each page; and that answering to the seconds of parallax is given in the intermediate part of the Table. The correction depending on the sun's or star's apparent altitude is placed at the bottom of the Table in each page.
As the size of the paper would not admit of the complete insertion of the auxiliary angle, except in the first vertical column of each page under or over 53!; therefore, in the eight following columns, it is only the excess of the auxiliary angle above 60 degrees that is given : hence, in taking out the auxiliary angle from those columns, it is always to be prefixed with 60 degrees.
The auxiliary angle is to be taken out of the Table, as thus :
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 the approximate auxiliary angle.
Enter the compartment of the “Proportional parts to seconds of parallax," abreast of the approximate auxiliary angle, with the tenths of seconds of the moon's horizontal parallax in the vertical column, and the units at the top; in the angle of meeting will be found a correction, which place under the approximate auxiliary angle; then enter the last or right-hand column of the page abreast of where the approximate auxiliary angle was found, or nearly so, and find the proportional part corresponding to the
odd minutes of altitude, which place under the former. To these three let the correction, at the bottom of the Table, answering to the sun's or star's apparent altitude, be applied, and the sum will be the correct auxiliary angle.
Let the moon's apparent altitude be 25:37:, the sun's apparent altitude 58:20", and the moon's horizontal parallax 59:47"; required the corresponding auxiliary angle?
Aux. angle ans, to moon's app. alt, 25.30', and hor. par. 59! is 60:13:47".
12 Proportional part to 7 minutes of altitude is
4 Correction corresponding to sun's app. alt. (58:20?) is :
Correction of the Auxiliary Angle when the Moon's Distance from a
Planet is observed.
The arguments of this Table are, a planet's apparent altitude in the left or right-hand column, and its horizontal parallax at top; in the angle of meeting stands the correction, which is always to be applied by addition to the auxiliary angle deduced from the preceding Table : hence, if the apparent altitude of a planet be 26 degrees, and its horizontal parallax 23 seconds, the correction of the auxiliary angle will be 6 seconds, additive.
This Table was calculated by a modification of the rule (page 43) for computing the correction of the auxiliary angle, answering to the sun's or star's apparent altitude; as thus :
To the logarithmic secant of the planet's apparent altitude, add the logarithmic cosine of its true altitude, and the constant logarithm 9.698850;* and the sum (abating 20 in the index) will be the logarithmic cosine of an arch ; the difference between which and 60 degrees will be the required correction.
* This is the log. cosine of 60 degrees diminished by .000120, the difference between the log. cosines of the true and apparent altitude of a fixed star between 30 and 90
Example Let the apparent altitude of a planet be 30 degrees, and its horizontal parallax 23 seconds ; required the correction of the auxiliary angle? Planet's apparent altitude 30: 0! 0 Log. secant 10.062469 Refrac. Table VIII 1.38
difference 1:18! Parallax, Table VI. 0.20 $
Const. log. 9.698850 True altitude of the planet
29:58:42" Loģ. cosine 9.937626
60: 0: 7"=Log. cosine 9.698945 60. 0.0
0: 0: 7"; which is the required
Error arising from a Deviation of one Minute in the Parallelism of
the Surfaces of the Central Mirror of the Circular Instrument of Reflection.
This Table contains the error of observation arising from a deviation of one minute in the parallelism of the surfaces of the central mirror of the reflecting circle, the axis of the telescope being supposed to make an angle of 80 degrees with the horizon mirror; it is very useful in finding the verification of the parallelism of the surfaces of the central mirror in the reflecting circle, or of the index glass in the sextant; as thus :
Let the instrument be carefully adjusted, and then take four or five observations of the angular distance between two well-defined objects, whose distance is not less than 100 degrees; the sum of these, divided by their number, will be the mean observation. Then,
Take out the central mirror, and turn it so that the edge which was before uppermost may now be downwards, or next the plane of the instrument; rectify its position, and take an equal number of observations of the angular distance between the same two objects, and find their mean, as before: now, half the difference between the mean of these and that of the former, will be the error of the mirror answering to the observed angle. If the first mean exceeds the second, the error is subtractive; otherwise additive: the mirror being in its first or natural position. Hence, if the
mean of the first set of observations be 115.0.40?, and that of the second 114:59:20", half their difference, viz., 1:20? • 2 = 40”, will be the error of the observed angle, and is subtractive; because the first mean angular distance, or that taken with the mirror in its natural position, is greater than the second, or that taken with the mirror inverted.
Having thus determined the error of the observed angle, that answering to any given angle may be readily computed by means of the present Table, as follows :
Enter the left-hand column of the Table with the angular distance, by which the error of the central mirror was determined, and take out the corresponding number from the adjoining column, or that marked "Observation to the right;" in the same manner take out the number answering to the given angle; then,
To the arithmetical complement of the proportional log. of the first number, add the proportional log. of the second, and the proportional log. of the observed error; the sum of these three logs., rejecting 10 from the index, will be the proportional log. of the error answering to such given angle.
Example. Having found the error arising from a defect of parallelism in the central mirror, åt an angle of 115 degrees, to be 40 seconds subtractive; required the error corresponding to an angle of 85 degrees? Obs. ang. 115 deg. opp. to which is 3:23". Arith. comp. prop. log.=8.2741 Given ang 85 deg.opp. to which is 1?15”
. Propor. log.
2. 1584 Observed error of central mirror 0.40 Propor. log.
Error of Observation arising from an Inclination of the Line of Collima
tion to the Plane of the Sextant, or to that of the circular Instrument of Reflection.
If the line of sight is not parallel to the planè of the instrument, the angle measured by such instrument will always be greater than the true angle. This Table contains the error arising from that cause, adapted to the most probable limits of the inclination of the line of collimation, and to any angle under 120 degrees : hence the arguments of the Table are, the observed angle in the left-hand column, and the inclination of the line