50. But if the star changes its declination, the greatest altitude

ceases to be the meridian altitude. Let h denote the hour angle of the star at the time of observation. Then if the star did not change its declination, and if B were the number of seconds given by Table XXXII for the diminution of altitude in one minute from the meridian passage, ho B would be the diminution of altitude in h minutes. But, since his small, the altitude, at this time, is increased by the change of declination; so that if A is the number of minutes by which the star changes its declination in one hour, that is, the number of seconds by which it changes its declination in one minute, h A will be the increase of altitude in the time of h, so that the altitude at the time h exceeds the meridian altitude by

If, then, h denotes the time of the greatest altitude, and h + 8 h a time which differs very slightly from the greatest altitude; the greatest altitude exceeds the altitude at the time h+8 h by the quantity

(h A — h2 B) — [ (h + 8 h) A − (h + 8 h)o B]
= 8 h [(− A + 2 B h) + B 8 h],

(453)

and 8 h can be supposed so small that B8 h may be insensible, and (453) becomes

8h (A2 Bh).

(454)

Now A+ 2 Bh cannot be negative, because h is supposed to correspond to the greatest altitude, and cannot be less than the altitude at the time h+8 h. Neither can— A + 2 B h be positive, for the altitude at the time h exceeds that at the time h—8 h by the quantity

8h (A+ 2 B h),

which, in this case, would be negative, and the altitude at the time — dh would exceed the greatest altitude. Since, then, − A + 2 B h can neither be greater nor less than zero, we must

have

or

and this value of h, substituted in (452), gives

for the excess of the greatest altitude above the meridian altitude.

51. If the observer were not at rest, his change of latitude will affect his observed greatest altitude in the same way in which it would be affected by an equal change in the declination of the star; so that the calculation of the correction on this account may be made by means of (455) and (456) precisely as in [B., p. 169.]

52. EXAMPLES.

1. An observer sailing N. N. W. 9 miles per hour, found by observation, the greatest central altitude of the moon bearing south, to be 54° 18'; what was the latitude, if the moon's declination was 6° 30' S., and her increase of declination per hour 16'.52?

2. An observer sailing south 12 miles per hour, found, by observation, the greatest central altitude of the moon bearing south, to be 25° 15'; what was the latitude, if the moon's declination was 1° 12′ N., and her increase of declination per hour 18'.5?

Ans. 66° 1' N.

19

CHAPTER V.

THE ECLIPTIC.

53. THE careful observation of the sun's motion shows this body to move nearly in the circumference of a great circle. This circle is called the ecliptic. [B., p. 48.]

54. The angle which the ecliptic makes with the equator is called the obliquity of the ecliptic.

55. The points, where the ecliptic intersects the equator, are called the equinoctial points; or the equinoxes. The point through which the sun ascends from the southern to the northern side of the equator, is called the vernal equinox; and the other equinox is called the autumnal equinox.

The points 90° distant from the ecliptic are called the solstitial points, or the solstices. [B., p. 49.]

56. The circumference of the ecliptic is divided into twelve equal parts, called signs, beginning with the vernal equinox, and proceeding with the sun from west to east.

The names of these signs are Aries (Y), Taurus (8), Gemini (□), Cancer (), Leo (N), Virgo (mg), Libra (~), Scorpio (m), Sagittarius (1), Capricornus (), Aquarius (m), Pisces (). The vernal equinox is therefore the first point, or beginning of Aries, and the autumnal equinox is the first point of Libra; the first six signs are north of the equator, and the last six south of the equator. The northern solstice is the first point of Cancer, and the southern solstice the first point of Capricorn. [B., p. 49.]

57. Secondary circles, drawn perpendicular to the ecliptic, are called circles of latitude.

The circle of latitude drawn through the equinoxes is called the equinoctial colure.

The circle of latitude drawn through the solstices is called the solstitial colure. [B., p. 49.]

Corollary. The solstitial colure is also a secondary to the equator, so that it passes through the poles of both the equator and the ecliptic.

58. Small circles, drawn parallel to the equator through the solstitial points, are called tropics.

The northern tropic is called the tropic of Cancer; the southern tropic the tropic of Capricorn.

Small circles, drawn at the same distance from the poles which the tropics are from the equator, are called polar circles.

"

The northern polar circle is called the arctic circle, the southern the antarctic.

59. The latitude of a star is its distance from the ecliptic measured upon the circle of latitude, which passes through the star. If the observer is supposed to be at the earth, the latitude is called geocentric latitude; but if he is at the sun, it is heliocentric latitude. [B., p. 49.]

60. The longitude of a star is the arc of the ecliptic contained between the circle of latitude drawn through the star and the vernal equinox. [B., p. 50.]

Corollary. The longitude and right ascension of the first point of Cancer are each equal to 6", and those of the first point of Capricorn are each equal to 18h.

61. The nonagesimal point of the ecliptic is the highest point at any time.

Corollary. The distance of the nonagesimal from the zenith is therefore equal to the distance of the zenith from the ecliptic, that is,

to the celestial latitude of the zenith; and the longitude of the nonagesimal is the celestial longitude of the zenith.

62. Problem. To find the latitude and longitude of a star, when its right ascension and declination are known.

Solution. Let P (fig. 35) be the north pole of the equator, Z the north pole of the ecliptic, and B the star. Then EQW will be the equator, NESW the ecliptic, and NPZS the solstitial colure, so that the point S is the southern solstice, and N the northern solstice. Now if the arc PB be produced to cut the equator at M, and ZB to cut the ecliptic at L; the angle ZPB is measured by the arc QM, that is, by the difference of the right ascensions of Q and M, or by the difference of the 's right ascension and 184; that is,

or

or

ZPB=184 — R. A. — 24a — (6a + R. A.)

= R. A. — 18* = (R. A. + 6a) — 24a

= 24 + R. A. — 18a — R. A. † 6a.

In the same way

PZB= NL = Long.-90°

or

=360° — (Long. — 90°)
(Long.

90°),

in which the first values of ZPB and PZB correspond to the star's being east of the solstitial colure; the second and third values to the star's being west of the colure. We also have

in which the declination and latitude are positive when north, and negative when south, and E has the same sign with R. A.—12a.

The present problem does not, then, differ from that of § 28, and if we put

A PC-90°,