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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 :: 1: tan. 20":5; and, therefore, the angle SP Q, or PSR, by which the axis of the telescope must deviate from the true direction of the star, must be 20" 5.
(332.) A similar reasoning will hold good when the direction of the earth's motion is not perpendicular to the visual ray. If SB be the true direction of the visual ray, and AC the position 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 D it matters not whether we use the sines or tangents). But we have, also, by trigonometry, BC: BA:: sine of B AC: sine of A CB or CBP, 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.
(333.) 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 scen if the earth were at rest.
(334.) 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 embracing 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.
(335.) When the body from which the visual ray emanates is itself in motion, an effect arises which is not properly speaking aberration, though it is usually treated under that head in astronomical books, and indeed confounded with it, to the production of some confusion in the mind of the student. The effect in question (which is independent of any theoretical views respecting the nature of light*) may be explained as follows. The ray by which we see any object is not that which 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 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 budy 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 ainount to above six thousandths of a second.
a 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 the total amount of its apparent misplacement. Its effect is to displace the body observed in a direction contrary to its apparent motion in the heavens. And it is a compound or aggregate effect consisting of two parts, one of which is the aberration, properly so called, resulting from the composition of the earth's motion with that of light, the other being what is not inaptly termed the Equation of light, being the allowance to be made for the time occupied by the light in traversing a variable space.
(336.) The complete Reduction, as it is called, of an astronomical observation consists in applying to the place of the observed heavenly body as read off on the instruments (supposed perfect and in perfect adjustment) five distinct and independent corrections, viz. those for refraction, parallax, aberration, precession, and nutation. Of these the correction for refraction enables us to declare what would have been the observed place, were there no atmosphere to displace it. That for parallax enables us to say from its place observed at the surface of the earth, where it would have been seen if observed from the centre. That for aberration, where it would have been observed from a motionless, instead of a moving station : while the corrections for precession and nutation refer it to fixed and determinate instead of constantly varying celestial circles. The great importance of these corrections, which pervade all astronomy, and have to be applied to every observation before it can be employed for any practical or theoretical purpose, renders this recapitulation far from superfluous.
(337.) Refraction has been already sufficiently explained, Art. 40. and it is only, therefore, necessary here to add that in its use as an astronomical correction its amount must be applied in a contrary sense to that in which it affects the observation ; a remark equally applicable to all other corrections.
(338.) The general nature of parallax or rather of parallactic motion has also been explained in Art. 80. But parallax in the uranographical sense of the word has a more technical meaning. It is understood to express that optical displacement of a body observed which is due to its being observed, not from that point which we have fixed upon as a conventional central station (from which we conceive the apparent motion would be more simple in its laws), but from some other station remote from such conventional centre: not from the centre of the earth, for instance, but from its surface: not from the centre of the sun (which, as we shall hereafter see, is for some purposes a preferable conventional station), but from that of the earth. In the former case this optical displacement is called the diurnal or geocentric parallax; in the latter the annual or heliocentric. In either case parallax is the correction to be applied to the apparent place of the heavenly body, as actually seen from the station of observation, to reduce it to its place as it would have been seen at that instant from the conventional station.
(339.) The diurnal or geocentric parallax at any place of the earth's surface is easily calculated if we know the distance of the body, and, vice versâ, if we know the diurnal parallax that distance may be calculated. For supposing S the object, C the centre of the earth, A the station of observation at its surface, and CAZ the direction of a perpendicular to the surface at A, then will the object be seen from A in the direction AS, and its apparent zenith dis. T tance will
be ZAS; whereas, if seen from the centre, it will appear in the direction CS, with an angular distance from the zenith of A equal to Z CS; so that ZAS-ZCS or ASC is the parallax. Now since by trigonometry CS:CA :: sin CAS = sin
ZAS: sin ASC, it follows that the sine of the parallax Radius of earth
x sin Z A S. Distance of body
(340.) The diurnal or geocentric parallax, therefore, at a given place, and for a given distance of the body observed, is proportional to the sine of its apparent zenith distance, and is, therefore, the greatest when the body is observed in the act of rising or setting, in which case its parallax is called its horizontal parallax, so that at any other zenith distance, parallax = horizontal parallax x sine of apparent zenith distance, and since ACS is always less than Z A S it appears that the application of the reduction or correction for parallax always acts in diminution of the apparent zenith distance or increase of the apparent altitude or distance from the Nadir, i.e. in a contrary sense to that for refraction.
(341.) In precisely the same manner as the geocentric or diurnal parallax refers itself to the zenith of the observer for its direction and quantitative rule, so the heliocentric or annual parallax refers itself for its law to the point in the heavens diametrically opposite to the place of the sun as seen from the earth. Applied as a correction, its effect takes place in a plane passing through the sun, the earth, and the observed body. Its effect is always to decrease its observed distance from that point or to increase its angular distance from the sun. And its sine is given by the relation, Distance of the observed body from the sun : distance of the earth from the sun :: sine of apparent angular distance of the body from the sun (or its apparent elongation): sine of heliocentric parallax. *
(342.) On a summary view of the whole of the uranographical corrections, they divide themselves into two classes, those which do, and those which do not, alter the apparent configurations of the heavenly bodies inter se. The former are real, the latter technical corrections. The real corrections are refraction, aberration and parallax. The technical are
This account of the law of heliocentric parallax is in anticipation of what follows in a subsequent chapter, and will be better understood by the student when somewhat farther advanced.