Parallax in latitude and longitude.

141. Corollary. Another process for computing & D may be obtained from (762). This equation gives

= P cos. N cos. A' cos. p + Pp cos. N sin. A'

= P cos. N cos. A' cos. p + P. P cos. A cos. N sin. A'

= P cos. N cos. A' cos. p +- P § D sin. A'.

(768)

Let

n' = P sin. A',

(769)

and (768) gives

(1 — n') & D = P cos. N cos. A' cos. p

and A and N" can be deduced by direct solution of the triangle ZHP, in which

ZPH = h + h, PH = PM 90° - D nearly

and A may be substituted for A' in determining the value of the small quantity n' by means of (769), and sec. ♪ D may be substituted for sec. p.

Apparent diameter.

142. Problem. To find the parallax in right ascension and declination.

Solution. Formulas (761-771) may be applied immediately to this case, by putting

B = the altitude of the equator the co-latitude,

D

the true declination,

D' the apparent declination,

h = the right ascension of the body diminished by that of the zenith the hour angle of the body.

D = the parallax in declination,

sh=the parallax in right ascension.

And formulas (761, 767, 771) correspond to those given by Woodhouse, in his method of calculating eclipses and occultations, in the Nautical Almanac for 1826. The mean values of sec. & D and sec. p are there substituted for them, which is 0.00006.

143. The apparent diameter of a heavenly body is the angle which its disc subtends.

144. Problem. To find the apparent semidiameter of a heavenly body.

Solution. Let O' (fig. 56) be the centre of the heavenly body, A the observer, and AT the tangent to the disc of the body. The angle TAO' is the apparent semidiameter. Let

Apparent diameter.

Hence, by (fig. 53), if A is the apparent altitude of the

which is also the semidiameter, as seen from the earth's cen

Ꭱ .

19.43537, (ar. co.) =0.5646, (781)

R

Augmentation of semidiameter.

so that formula (775) agrees with [B. p. 443. No. 10 of the Rule].

146. Corollary. If so is the augmentation of the semidiameter for the altitude A, we have, by (776),

or, in order to express & and P in seconds,

Now for the mean horizontal parallax of 57' 30", we have

agreeing very nearly with the explanation to Table XV of the Navigator.

147. Corollary. The augmentation can also be calculated without determining the altitude. Thus, from (774)

cos. D sin. h

P. cos. h. sin. B sin. (h+h), cos.(D-8D)

Now the latitude of the moon is so small, that, in the first

+

-1

cos. D

H= } . P. [sin. (B+h) + sin. (B–h)]

(794]

H'= 2 (tang. D . § D + cos. ♪ D.

− 1),

(795)

and formulas (793 to 795) agree with the method of calculating the augmentation of the semidiameter given in Table XLIV of the Navigator. The three first parts of this table are calculated for the value of Σ,