The Polar Circles are two small circles parallel to the equator, and at a distance from its poles equal to the obliquity of the ecliptic. That about the north pole is called the arctic circle, and that about the south pole, the antarctic. Thus pq is the arctic and p'q' the antarctic circle.

Circles corresponding to the tropics and polar circles, and bearing the same names are conceived to be drawn on the earth's surface, dividing it into five portions called zones. The zone between the tropics is called the torrid zone; the two between the tropics and polar circles are called the temperate zones and the two within the polar circles are called the frigid zones.

115. The Right Ascension of a body is the arc of the equator intercepted, to the east, between the vernal equinox and a declination circle passing through the body. Thus EG is the right ascension of the star S.

The right ascension and declination (27) of a body, designate its situation in reference to the equator.

116. A Circle of Latitude is through the poles of the ecliptic. a circle of latitude.

any great circle passing

The arc pSH is part of

117. The Longitude of a body is the arc of the ecliptic intercepted, to the east, between the vernal equinox and a circle of latitude passing through the body.

The Latitude of a body is the arc of a circle of latitude intercepted between the body and the ecliptic. The latitude is north or south, according as the body is on the north or south side of the ecliptic. Thus, EH is the longitude and HS the latitude of the star S, north.

The longitude and latitude of a body designate its place in reference to the ecliptic.

PROBLEMS.

118. To find the obliquity of the ecliptic. The obliquity of the ecliptic may be found from the equation, tang dď =

tang E sin Eď, (106), in which dd and Ed are known from observation, and the angle E is the obliquity of the ecliptic. This gives,

It may however be more accurately obtained from the sun's declination, found for several days at noon (102), about the time of either solstice. From these declinations, the value of CQ, the greatest declination, may be deduced by interpolation; and this expresses the obliquity of the ecliptic (110).

The obliquity is subject to some slight changes that will be noticed in the next chapter.

119. To change the right ascension and declination of a body into longitude and latitude or the contrary.

Let S be the body and let E and S be joined by an arc of a great circle. Put angle QEC ecliptic, which is supposed to be known, A

=

=

obliquity of the EG, the

right ascension, DGS, the declination, L = EH, the longitude, λ = HS,the latitude, N

=

the angle GES, and

Then for the change from right ascension and declination, to longitude and latitude, we have from the triangles EGS and EHS, (App. 48 and 49).

tang tang (N) sin L.

For the change from longitude and latitude to right ascension and declination, we have in like manner,

tang and tang ES

=

tang L cos N

Hence,

;

When the declination or latitude is south it must be taken negative. The subsiduary arc N or N' may always be taken affirmative and less than 180°. When one of the quantities L and R, is in either the fourth or first quadrant, it is evident the other must be in one of these two; and when one of them is in either the second or third quadrant, the other must be in one of these. With attention to these remarks and to the trigonometrical rules for the signs of the quantities, the preceding formulæ are applicable whatever be the situation of the body.

For the sun or any point in the ecliptic as S', we evidently have,

120. Positions of the fixed stars.

When EG, the right ascension of one star S has been obtained (106), the right ascension of any other body may be found from the observed interval in sidereal time, between its passage over the meridian and that of the star. This interval added to the right ascension of the star, expressed in time, or subtracted from it, according as the passage of the body is later or earlier than that of the star, will evidently give its right ascension in time. The method of finding the polar distance or declination has been already given (102).

When the right ascensions and declinations of the stars have been found from observations, their longitudes and latitudes may, if required, be computed by the last article.

121. Constellations. The ancients, in order to distinguish the various groups of stars, imagined figures of men, animals and other objects, to be drawn around them in the concave surface of the celestial sphere. The group of stars

contained within the contour of any one of these imaginary figures is called a Constellation. Each constellation bears the name of the figure which limits it.

The number of constellations formed by the ancients is 48. To these about 40 have since been added; some of them being small constellations, formed of stars not included in the ancient constellations, but most of them are in that part of the southern hemisphere not visible to the ancient observers. Twelve of the constellations follow one another along the ecliptic, and bear the same names as its signs. These are called zodiacal constellations.

122.

Stars of a constellation. The stars of a constellation are distinguished from one another by the letters of the Greek alphabet, which are applied to them according to their apparent relative size or brightness. The principal star in the constellation is named, the second 8, the third », and thus on. When the number of stars in a constellation exceeds the number of letters in the Greek alphabet, as it generally does, the remainder are designated by the letters of the Roman alphabet or by numbers. The expression Lyra, denotes the star a in the constellation Lyra, a harp; and so of others.

α

Some of the stars have particular names, as Sirius, Aldebaran, Arcturus, &c.

123. Definition. A Catalogue of fixed stars is a table containing a list of stars with their right ascension and declinations, or their longitudes and latitudes.

The first catalogue was formed by Hipparchus about 130 years prior to the Christian era; and contained the positions of nearly 1000 stars. Various catalogues have since been formed; some of them containing the situations of many thousands of stars, most of which are only visible by the aid of a telescope.

CHAPTER IX.

PRECESSION OF THE EQUINOXES-ABERRATION-NUTATION.

124. Position of the ecliptic and motion of the equinoxes. From comparisons of catalogues of the stars, formed at different times it is found that the latitudes of the stars continue always nearly the same. Hence the position of the ecliptic among the stars must be fixed, or nearly so.

But it is found from these comparisons that the longitudes of the stars are continually increasing at the rate of about 50" in a year. This increase of longitude is common to all the stars, and, except for a few, is the same for each star. It cannot therefore be reasonably imputed to motions in the stars themselves. Hence it follows that the vernal equinox, the point from which longitude is reckoned, must have a backward or retrograde motion along the ecliptic, equal to the increase in the longitudes of the star. Let ECFD, Fig 20, be the ecliptic, p and p' its poles, E the place of the vernal equinox at any time, and E' its place at some subsequent time, it having during the intermediate time, retrograded along the ecliptic through the arc EE'. Then must the longitude of any star S, be changed during this interval of time from EH to E'H; being increased by the quantity EE'.

As the autumnal equinox is always directly opposite to the vernal equinox, it must have the same motion.

125. Definition. The Precession of the Equinoxes is the retrograde motion which they have along the ecliptic. It is 50".2 in a year.

126. The poles of the equator revolve with retrograde motions in small circles around the poles of the ecliptic, at distances equal to the obliquity of the ecliptic.

As the ecliptic remains in a fixed position or nearly so