reached C, the tube should have been carried into the position R S, it is evident that the ball would, throughout its whole descent, be found in the axis of the tube; and a spectator, referring to the tube the motion of the ball, and carried along with the former, unconscious of its motion, would fancy that the ball had been moving in the inclined direction RS of the tube's axis.

(277.) Our eyes and telescopes are such tubes. In whatever manner we consider light, whether as an advancing wave in a motionless ether, or a shower of atoms traversing space, if in the interval between the rays traversing the object glass of the one or the cornea of the other (at which moment they acquire that convergence which directs them to a certain point in fixed space), the cross wires of the one or the retina of the other be slipped aside, the point of convergence (which remains unchanged) will no longer correspond to the intersection of the wires or the central point of our visual area. The object then will appear displaced; and the amount of this displacement is aberration.

(278.) The earth is moving through space with a velocity of about 19 miles per second, in an elliptic path round the sun, and is therefore changing the direction of its motion at every instant. Light travels with a velocity of 192,000 miles per second, which, although much greater than that of the earth, is yet not infinitely so. Time is occupied by it in traversing any space, and in that time the earth describes a space which is to the former as 19 to 192,000, or as the tangent of 20"-5 to radius. Suppose now A P S to represent a ray of light from a star at A, and let the tube PQ be that of a telescope so inclined forward that the focus formed by its object glass shall be received upon its cross wire, it is evident from what has been said, that the inclination of the tube must be such as to make PS: SQ:: velocity of light: velocity of the earth, :: tan. 20"-5: 1; and, therefore, the angle SPQ, or PSR, by which the axis of the telescope must deviate from the true direction of the star, must be 20" 5.

CHAP. IV.

CORRECTION FOR ABERRATION.

179

(279.) A similar reasoning will hold good when the direction of the earth's motion is not perpendicular to

B

the visual ray. If SB be the true direction of the visual ray, and A C the posi.. tion in which the telescope requires to be held in the apparent direction, we must still have the proportion BC: BA:: velocity of

light: velocity of the earth :: rad.: sine of 20′′-5 (for in such small angles it matters not whether we use the sines or tangents). But we have, also, by trigonometry, BC : BA:: sine of BAC : sine of AC B or C B D, which last is the apparent displacement caused by aberration. Thus it appears that the sine of the aberration, or (since the angle is extremely small) the aberration itself, is proportional to the sine of the angle made by the earth's motion in space with the visual ray, and is therefore a maximum when the line of sight is perpendicular to the direction of the earth's motion.

(280.) The uranographical effect of aberration, then, is to distort the aspect of the heavens, causing all the stars to crowd as it were directly towards that point in the heavens which is the vanishing point of all lines parallel to that in which the earth is for the moment moving. As the earth moves round the sun in the plane of the ecliptic, this point must lie in that plane, 90° in advance of the earth's longitude, or 90° behind the sun's, and shifts of course continually, describing the circumference of the ecliptic in a year. It is easy to demonstrate that the effect on each particular star will be to make it apparently describe a small ellipse in the heavens, having for its centre the point in which the star would be seen if the earth were at rest.

(281.) Aberration then affects the apparent right ascensions and declinations of all the stars, and that by quantities easily calculable. The formulæ most convenient for that purpose, and which, systematically embra、

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 velo. city 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,

CHAP. IV.

URANOGRAPHICAL PROBLEMS.

181

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 PS being also given, and the co-latitude ZP 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 Z PE, 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

E

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