cing at the same time the corrections for precession and nutation, enable the observer, with the utmost readiness, to disencumber his observations of right ascension and declination of their influence, have been constructed by Prof. Bessel, and tabulated in the appendix to the first volume of the Transactions of the Astronomical Society, where they will be found accompanied with an extensive catalogue of the places, for 1830, of the principal fixed stars, one of the most useful and best arranged works of the kind which has ever appeared.

(282.) When the body from which the visual ray emanates is, itself, in motion, the best way of conceiving the effect of aberration (independently of theoretical views respecting the nature of light)* is as follows. The ray by which we see any object is not that which it emits at the moment we look at it, but that which it did emit some time before, viz. the time occupied by light in traversing the interval which separates it from us. The aberration of such a body then arising from the earth's velocity must be applied as a correction, not to the line joining the earth's place at the moment of observation with that occupied by the body at the same moment, but at that antecedent instant when the ray quitted it. Hence it is easy to derive the rule given by astronomical writers for the case of a moving object. From the known laws of its motion and the earth's, calculate its apparent or relative angular motion in the time taken by light to traverse its distance from the earth. This is its aberration, and its effect is to displace it in a direction contrary to its apparent relative motion among the stars.

We shall conclude this chapter with a few urano

The results of the undulatory and corpuscular theories of light, in the matter of aberration are, in the main, the same. We say in the main. There is, however, a minute difference even of numerical results. In the undulatory doctrine, the propagation of light takes place with equal velocity in all directions whether the luminary be at rest or in motion. In the corpuscular, with an excess of velocity in the direction of the motion over that in the contrary equal to twice the velocity of the body's motion. In the cases, then, of a body moving with equal velocity directly to and directly from the earth, the aberrations will be alike on the undulatory, but different on the corpuscular hypothesis. The utmost difference which can arise from this cause in our system cannot amount to above six thousandths of a second.

graphical problems of frequent practical occurrence, which may be resolved by the rules of spherical trigonometry.

(283.) Of the following five quantities, given any three, to find one or both the others.

1st, The latitude of the place; 2d, the declination of an object; 3d, its hour angle east or west from the meridian; 4th, its altitude; 5th, its azimuth.

In the figure of art. 94. P is the pole, Z the zenith, and S the star; and the five quantities above mentioned, or their complements, constitute the sides and angles of the spherical triangle PZS; PZ being the co-latitude, PS the co-declination or polar distance; SPZ the hour angle; PS the co-altitude or zenith distance; and PZS the azimuth. By the solution of this spherical triangle then, all problems involving the relations between these quantities may be resolved.

(284.) For example, suppose the time of rising or setting of the sun or of a star were required, having given its right ascension and polar distance. The star rises when apparently on the horizon, or really about 34' below it (owing to refraction), so that, at the moment of its apparent rising, its zenith distance is 90° 34' ZS. = Its polar distance P S being also given, and the co-latitude Z P of the place, we have given the three sides of the triangle, to find the hour angle Z P S, which, being known, is to be added to or subtracted from the star's right ascension, to give the sidereal time of setting or rising, which, if we please, may be converted into solar time by the proper rules and tables.

(285.) As another example of the same triangle, we may propose to find the local sidereal time, and the latitude of the place of observation, by observing equal altitudes of the same star east and west of the meridian, and noting the interval of the observations in sidereal time.

The hour angles corresponding to equal altitudes of a fixed star being equal, the hour angle east or west

will be measured by half the observed interval of the observations. In our triangle, then, we have given this hour angle ZPS, the polar distance PS of the star, and ZS, its co-altitude at the moment of observation. Hence we may find P Z, the co-latitude of the place. Moreover, the hour angle of the star being known, and also its right ascension, the point of the equinoctial is known, which is on the meridian at the moment of observation; and, therefore, the local sidereal time at that moment. This is a very useful observation for determining the latitude and time at an unknown station.

(286.) It is often of use to know the situation of the ecliptic in the visible heavens at any instant; that is to say, the points where it cuts the horizon, and the altitude of its highest point, or, as it is sometimes called, the nonagesimal point of the ecliptic, as well as the longitude of this point on the ecliptic itself from the equinox. These, and all questions referable to the same data and quæsita, are resolved by the spherical triangle ZPE, formed by the zenith Z (considered as the pole of the horizon), the pole of the equinoctial P, and the pole of the ecliptic E. The sidereal time being given, and also

the right ascension of the pole of the ecliptic (which is always the same, viz. 18h 0m 0s), the hour angle Z PE of that point is known. Then, in this triangle we have

given P Z, the co-latitude; PE, the polar distance of the pole of the ecliptic, 23° 28′, and the angle Z PE; from which we may find, 1st, the side Z E, which is easily seen to be equal to the altitude of the nonagesimal point sought; and, 2dly, the angle P ZE, which is the azimuth of the pole of the ecliptic, and which, therefore, being added to and subtracted from 90°, gives the azimuths of the eastern and western intersections of the ecliptic with the horizon. Lastly, the longitude of the nonagesimal point may be had, by calculating in the same triangle the angle PEZ, which is its complement.

(287.) The angle of situation of a star is the angle inIcluded at the star between circles of latitude and of declination passing through it. To determine it in any proposed case, we must resolve the triangle PSE, in which are given PS, PE, and the angle SP E, which is the difference between the star's right ascension and 18 hours; from which it is easy to find the angle PSE required. This angle is of use in many enquiries in physical astronomy. It is called in most books on astronomy the angle of position; but the latter expression has become otherwise, and more conveniently, appropriated.

(288.) From these instances, the manner of treating such questions in uranography as depend on spherical trigonometry will be evident, and will, for the most part, offer little difficulty, if the student will bear in mind, as a practical maxim, rather to consider the poles of the great circles which his question refers to, than the circles themselves.

CHAP. V.

OF THE SUN'S MOTION.

APPARENT MOTION OF THE SUN NOT UNIFORM. ITS APPARENT DIAMETER ALSO VARIABLE. VARIATION OF ITS DISTANCE CONCLUDED. ITS APPARENT ORBIT AN ELLIPSE ABOUT THE FOCUS. LAW OF THE ANGULAR VELOCITY. — EQUABLE DESCRIPTION OF AREAS. PARALLAX OF THE SUN. ITS DISTANCE AND MAGNITUde. COPERNICAN EXPLANATION OF THE SUN'S APPARENT MOTION. - PARALLELISM OF THE EARTH'S AXIS. THE SEASONS. -HEAT RECEIVED FROM THE SUN IN DIFFERENT PARTS OF THE ORBIT.

(289.) In the foregoing chapters, it has been shown that the apparent path of the sun is a great circle of the sphere, which it performs in a period of one sidereal year. From this it follows, that the line joining the earth and sun lies constantly in one plane; and that, therefore, whatever be the real motion from which this apparent motion arises, it must be confined to one plane, which is called the plane of the ecliptic.

(290.) We have already seen (art. 118.) that the sun's motion in right ascension among the stars is not uniform. This is partly accounted for by the obliquity of the ecliptic, in consequence of which equal variations in longitude do not correspond to equal changes of right ascension. But if we observe the place of the sun daily throughout the year, by the transit and circle, and from these calculate the longitude for each day, it will still be found that, even in its own proper path, its apparent angular motion is far from uniform. The change of longitude in twenty-four mean solar hours averages 0° 59' 8"-33; but about the 31st of December it amounts to 1° 1' 9"9, and about the 1st of July is only 0° 57′ 11′′-5. Such are the extreme limits, and such the mean value of the sun's apparent angular velocity in its annual orbit.

(291.) This variation of its angular velocity is accompanied with a corresponding change of its distance from