PROBLEM XVII.

To find the Apparent Longitude and Latitude, as affected by Parallax, and the Augmented Semi-diameter of the Moon; the Moon's True Longitude, Latitude, Horizontal Semi-diameter, and Equatorial Parallax, and the Longitude and Altitude of the Nonagesimal Degree of the Ecliptic, being given. We have for the resolution of this Problem the following formulæ :

log. x = log.P+log. cos. har.co.log. cos.λ—10.. (1);

c log. xlog. tang h log. u = c + log. sin K 10

10

...

· (2); (3);

log. u'elog. sin (Ku) - 10. . .

(4);

log. p = c + log. sin (K+u') - 10 . . . (5) ;

Appar. long.

log. tang

log. v

...

= log. p + ar. co. log. cos. λ + ar. co. log. u + log. sin (x) - 10 ... (7);

=

log. Plog. cos. h + log. cos. x 10... (8);

log. z = = log. v + log. tang h + log. tang >' + log. cos.
(K + p) - 30 . . . (9);
} ·

Appar. lat. = true lat. T (11);

log. R' = log. par. co. log. cos. λ+ar. co. log. u + log. cos. X' + log. R 10 . (12);

in which,

P = the Reduced Parallax of the Moon;

h

=

λ =

the Altitude of the Nonagesimal;

the True Latitude of the Moon (minus when south); K = the Longitude of the Moon, minus the longitude of the Nonagesimal ;

p = the required Parallax in Longitude;

λ= the approximate Apparent Latitude of the Moon;

<= the required Parallax in Latitude;

T

R' the True Semi-diameter of the Moon;

=

R' the Augmented Semi-diameter of the Moon;

x, u, u', v, z, are auxiliary arcs.

Formulæ (1), (2), (3), (4), and (5), being resolved in succession, we derive the apparent longitude from formula (6); then the apparent latitude from equations (7), (8), (9), (10), (11); and lastly, the augmented semi-diameter from equation 12.

Attention

The latitude of the moon must be affected with the negative sign when south; and the apparent latitude will be south when it comes out negative. In performing the operations, it is to be remembered that the cosine of a negative arc has the same sign as the cosine of a positive arc of an equal number of degrees; but that the sine or tangent of a negative arc has the opposite sign from the sine or tangent of an equal positive arc. must also be paid to the signs in the addition and subtraction of arcs. Thus, two arcs affected with essential signs, which are to be added to each other, are to be added arithmetically, when they have like signs, but subtracted if they have unlike signs; and when one arc is to be taken from another, its sign is to be changed, and the two united according to their signs. An arithmetical sum, when taken, will have the same sign as each of the arcs; and an arithmetical difference the same sign as the greater

arc.

The use of negative arcs may be avoided, though the calculation would be somewhat longer, by using the true polar distance d, and the approximate apparent polar distance d', in place of a and X', substituting sin d for cos. λ, cos. (d+x) for sin (λ — x), sin d' for cos. X', log. co-tang d' for log. tang '; and .observing that P is to be subtracted from the true longitude in case the longitude of the nonagesimal exceeds the longitude of the moon ; that z, when it comes out negative, is to be added to v, which is always positive to the north of the tropic, otherwise subtracted; and that the parallax in latitude is to be applied according to its sign to the true polar distance.

In seeking for the logarithms of the trigonometrical lines, it will be sufficient to take those answering to the nearest tens of seconds.

Note 1. When great accuracy is not desired, u' may be taken

for

P, from which it can never differ more than a fraction of a second.

2. In solar eclipses, the moon's latitude is very small, and formula (7) may be changed into the following,

log.λ = log. par.co.log. cos.λ+ar.co.log.u+log. (λ — x) — 10 and cos. >' omitted in formula (12) without material error.

Formulæ (8), (9), (10), and (11), may also now be dispensed with, unless very great precision is desired, and the value of x given by the above formula taken for the apparent latitude.

It is to be observed also, that in eclipses of the sun P is taken equal to the reduced parallax of the moon minus the sun's horizontal parallax. By this the parallax of the sun in longitude and latitude is referred to the moon, and the relative apparent places of the sun and moon are correctly obtained, without the necessity of a separate computation of the sun's parallax in longitude and latitude.

Exam. 1. About the time of the middle of the occultation of the star Antares, on the 10th of May, 1838, the moon's longitude, by the Connaissance des Tems, was 247° 37' 6".7; latitude 4° 14' 14".7 S.; semi-diameter 15' 24".2; and equatorial parallax 56' 31".7; and the longitude of the nonagesimal at New York was 200° 12' 23"; the altitude 37° 0' 34"; required the apparent longitude and latitude, and the augmented semi-diameter of the moon at New York, at the time in question.

Exam. 2. About the middle of the eclipse of the sun on the 18th of September, 1838, the moon's longitude was 175° 29 19".0, latitude 47' 47".5, equatorial parallax 53' 53".5, and semidiameter 14' 41".1; and the longitude of the nonagesimal at New York was 216° 20' 50", the altitude 32° 15' 48" required the apparent longitude and latitude, and the augmented semidiameter of the moon.