Let a, b, c, d, e, Plate XXXIX. Fig. 3. be the path of a particle of light emitted by the star S, and let 1m be a telescope carried along with the earth in its annual motion in the orbit AB. In order that an observer may see the star S, the telescope lnz must be inclined to the direction of the light which comes from the star, and must be pointed towards s, so that the %tar. will appear • • at s, instead of at S, having Ss for its aberration, or the difference between its true and apparent place. Let us at first suppose, that the velocity of the earth in its orbit is the same as the velocity of light ; that the telescope is in the position lin, when the particle of light is at a; and that while the particle advances from a to b, from b to c, from c to d, and from d to c, the telescope advances through the equal spaces 1, 2 ; 2, 3; S, 4; 4, 5. When the telescope has reached the position 2n, the particle will have arrived at b, and by means of it the star will be visible in the telescope. When the telescope has arrived at the position 3o, the particle of light will be, at c, still visible in the telescope ; and when the telescope has taken succes sively the positions 412, 5q, the particle will be at d and c successively ; so that the partLele has actually moved along the axis of the tube without touching its sides, and there fore the star must have been visible in the direction of the tube's axis, or at s. In the present case, the velocity of light having been assumed equal to that of the earth's annual motion, the inclination of the tube must be 45°, and the aberration of the star 45° ; but as the real velo city of light is nearly 10,313 times greater than that of the earth, the inclination of the tube must be only 20", to allow the light to pass through it ; and therefore the aberration of the star Ss will be only 20". See a differ ent explanation of this phenomenon under the article ABERRATION.
If the light of the star S, Plate XXXIX. Fig. 4. moves through the space OA, in the time that the earth de scribes the portion AB of its orbit at right angles to OA, the aberration will be the angle BOA, which will be equal to the tangent of the velocity of the earth in its orbit, the velocity of light being radius. If the path of the earth is oblique to the motion of the star's light, as AC, then COA will be the aberration ; and when the di rection of the earth's motion is AD, opposite to OA, or Ad, in the same direction with OA, the aberration will be nothing. In all these cases, the sine of the ab erration will vary as the sine of the inclination of the path of the earth to the path of the star's light.
It is obvious, from the preceding explanation, that the aberration is always in the direction in which the earth is moving. When the earth is moving in the di rection I, 2, 3, the aberration will be in the direction Ss, Plate XXXIX. Fig. 3.; when the earth is moving from B to C at right angles to its former direction, the abberration will be St ; when the earth is moving from C to D, the aberration will be ; and when it is de scribing the part DA of its orbit, the aberration will be STU : But Ss, St, Sr, SW are each 20", therefore the star will describe a small circle in the heavens 40" in dia meter; the star being always 90° farther advanced in its small circle s t v so than the earth is in its own orbit.
We have now supposed that the star is situated in the pole of the ecliptic, or, what is the same thing, that the path described by the light of the star is at right angles to the motion of the earth, in which case the aberration will always be exactly 20". If the star is situated in the plane of the ecliptic at P, and if the earth is at E, mo ving in the direction EG at right angles to EP, the ab erration will be 20" as before, and the star will appear at q. While the earth is moving from E to N, its path is gradually becoming more oblique to the rays pro ceeding from the star, and therefore the aberrration will be less, and the star will appear nearer its true place P till the earth arrive at N, when the aberration will be nothing, the.light of the star and the earth moving in the same direction. In the progress of the earth from N to F, the aberration will gradually increase in the opposite direction Pr; and when the earth reaches F, it will again be 20", or Pr, the light of the star being now at right angles to the earth's path. In the progress of the from F to AI, it will again diminish, and vanish at M; and from Al to C, the aberration will again increase, and reach its maximum at E. Hence it follows, that if the star be in the plane of the ecliptic, its aberration is a maximum when the star is in opposition and conjunc tion with the sun, and that it vanishes when the star is in the quadratures. In the intermediate points between. the quadratures and the conjunctions and oppositions, the aberration is proportional to the sine of the star's distance front the qAdratures. When the star is in the pole of the ecliptic, or in the plane of the ecliptic, the aberration is always in the direction of the earth's mo tion, and therefore affects only the longitude of the star, and consequently its right ascension and declination. When the star, however, is above or below the ecliptic, its latitude will also be affected ; but in every case, the aberration will be greatest when the star is in its oppo sitions and conjunctions, and least in the quadratures. The aberration, however, will not vanish in the quadra tures, as the earth and the star's light can never move either exactly in the same, or exactly in the opposite direction, excepting in the case which has been already noticed. Tables for finding the aberration of the stars will be found among the Tables at the end of this article.
The places of the planets are likewise affected by the motion of the earth, combined with the motion of light. Thus if 0, Plate. XXXIX. Fig. 4. be the place of the planet, and if BA be the portion of the earth's orbit which it describes in the time that the light of the pla net moves from 0 to A, or rather the relative motion of the earth and planet, then when the earth has reached b, the light of the planet will be at d, and when the earth has reached A the light will be at A; so that Bo or bd will be the direction in which the planet is seen; for if we suppose Bo a telescope carried along with the earth, the light of the planet will, in every position of the tube, be found in its axis. Hence the aberration of a planet is always equal to of the earth during the time that light employs to pass from the planet to the earth.