KP is the obliquity of the ecliptic, K X the co-latitude (or complement of latitude), and the angle P K X the co-longitude of the object X. These are the data of our question, of which the first is constant, and the two latter are varied by the effect of precession and nutation; and their variations (considering the minuteness of the latter effect generally, and the small number of years in comparison of the whole period of 25,868, for which we ever require to estimate the effect of the former,) are of that order which may be regarded as infinitesimal in geometry, and treated as such without fear of error. The whole question, then, is reduced to this: In a spherical triangle K P X, in which one side K X is constant, and an angle K, and

adjacent side K P vary by given infinitesimal changes of the position of P: required the changes thence arising in the other side PX, and the angle KPX? This is a very simple and easy problem of spherical geometry, and being resolved, it gives at once the reductions we are seeking; for PX being the polar distance of the object, and the angle K P X its right ascension pius 90°, their variations are the very quantities we seek. It only remains, then, to express in proper form the amount of the

precession and nutation in longitude and latitude, when their amount in right ascension and declination will immediately be obtained.

(273.) The precession in latitude is zero, since the latitudes of objects are not changed by it: that in longitude is a quantity proportional to the time at the rate of 50"-10 per annum. With regard to the nutation in longitude and latitude, these are no other than the abscissa and ordinate of the little ellipse in which the pole moves. The law of its motion, however, therein, cannot be understood till the reader has been made acquainted with the principal features of the moon's motion on which it depends. See Chap. XI.

(274.) Another consequence of what has been shown respecting precession and nutation is, that sidereal time, as astronomers use it, i. e. as reckoned from the transit of the equinoctial point, is, not a mean or uniformly flowing quantity, being affected by nutation; and, moreover, that so reckoned, even when cleared of the periodical fluctuation of nutation, it does not strictly correspond to the earth's diurnal rotation. As the sun loses one day in the year on the stars, by its direct motion in longitude; so the equinox gains one day in 25,868 years on them by its retrogradation. We ought, therefore, as carefully to distinguish between mean and apparent sidereal as between mean and apparent solar time.

(275.) Neither precession nor nutation change the apparent places of celestial objects inter se. We see them, so far as these causes go, as they are, though from a station more or less unstable, as we see distant land objects correctly formed, though appearing to rise and fall when viewed from the heaving deck of a ship in the act of pitching and rolling. But there is an optical cause, independent of refraction or of perspective, which displaces them one among the other, and causes us to view the heavens under an aspect always to a certain slight extent false; and whose influence must be estimated and allowed for before we can obtain a precise

knowledge of the place of any object. This cause is what is called the aberration of light; a singular and surprising effect arising from this, that we occupy a station not at rest but in rapid motion; and that the apparent directions of the rays of light are not the same to a spectator in motion as to one at rest. As the estimation of its effect belongs to uranography, we must explain it here, though, in so doing, we must anticipate some of the results to be detailed in subsequent chapters.

(276.) Suppose a shower of rain to fall perpendicu larly in a dead calm; a person exposed to the shower, who should stand quite still and upright, would receive the drops on his hat, which would thus shelter him, but if he ran forward in any direction they would strike him in the face. The effect would be the same as if he remained still, and a wind should arise of the same velocity, and drift them against him. Suppose a ball let fall from a point A above a horizontal line E F, and that at B were placed to receive it the open mouth of

A:

F

P

B

E

an inclined hollow tube PQ; if the tube were held immoveable the ball would strike on its lower side, but if the tube were carried forward in the direction E F, with a velocity properly adjusted at every instant to that of the ball, while preserving its inclination to the horizon, so that when the ball in its natural descent

N

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 R S 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 PS R, by which the axis of the telescope must deviate from the true direction of the star, must be 20" 5.

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

C

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 B A C : sine of A C 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