tion. Corresponding variations are found to depend on the place of the sun, and these constitute the solar nutation. When the position of the pole P' in the ellipse of nutation is determined for any given time, the effects of nutation on the right ascension and polar distance of any particular star may be determined by spherical trigonometry in a manner exactly similar to that which has been described in finding the effects of the general precession. C 241. Thus, let p be the pole of the ecliptic, and P the mean place of the pole of the equator at a given epoch: also, agreeably to the hypothesis of Bradley, let it be supposed that ABCD in which the true pole p' is situated is an ellipse, and imagine the circle Abcd to be described about it. The pole P' being at a when the moon's ascending node is at Q (fig. to art. 239.), and at B when the node is at N, and so on; let the angle APE be equal to the distance of the node from Q in a direction contrary to the order of the signs (that is, let APE be equal to 360° minus the longitude of the node): then drawing EF perpendicular to AC, the place of P' may be conceived to be the intersection of this line with the periphery of the ellipse. Now, by conic sections, AP: BP:: EF : P′F, and by trigonometry, BP EPT A D EF PF: tan. EPA tan. P'PA. But, from the best observations, AP, called the constant of nutation, =9".239 and BP-7".18 (Mem. Astr. Soc., vol. xii.); and putting N for the longitude of the moon's node, tan. EPA = tan. N, also cos. EPA cos. N; PE PP sec. EPA sec. P'PA, or :: cos. P'PA: cos. EPA; Then, in the spherical triangle PPP' (art. 61.), sin. APP' : sin. P'pP:: sin. P'p: sin. PP'; or, putting Pp (equal to the obliquity of the ecliptic) for P'p and representing it by 0; also putting small arcs or angles for their sines, sin. APP' P'pP:: sin. : PP'; Substituting the above value of PP', we have 9".239 cos. N sin. APP' 9.239 cos. N tan. APP'. And again, substituting the above value of tan. APP', we have 9".239 cos. N 7.18 sin. tan. N, 9".239 which, after reduction, becomes - 18".03 sin. N. This is called the lunar equation of precession, or the lunar equation of the equinoctial points in longitude. From the value of PP' above, we have PF ( = PP′ cos. P'PA) = 9′′.239 cos. N. This is called the lunar equation of obliquity. 242. By processes exactly similar to those which were employed for finding the precession in right ascension and declination, the effects of nutation in those directions may be found. For let rc represent part of the trace of the ecliptic, of which Р is the pole; R the trace of part of the mean equator whose pole is P, and ER' the trace of part of the true equator at the given time; and, for this time, let N represent the longitude of the moon's node, let P' be the pole of the equator, and let s be the place of a star. The figure being completed as before (art. 226.), the arc E is equal to the angle Pp P', and represents the lunar equation of the equinoctial points in longitude: then, the angle at E being represented by 0, the obliquity; and E t p ရာ P P A m S C R R/ rt being let fall perpendicularly on ER', we have (Pl. Trigo., art. 56.) EY cos. Et; or, since EV = P'pp, This is called the equation of the equinoctial points in right ascension, in seconds of a degree; or when divided by 15, it expresses that equation in time. 243. Let fall P'r perpendicularly on PS; then RPS) = P'PR − (− a) (a representing rm the right ascension of the star): 2 Also Pr PP' cos. P'Pr PP' sin. (P′PR + a) PP' (sin. P'PR cos. a+ cos. P′PR sin. a); then, for PP' sin. P'PR substituting P'pP sin. 0 (art. 241), or 18.03 sin. N sin. 0; and for PP cos. P'PR substituting 9.239 cos. N, we get Pr9".239 cos. N sin. a — 18".03 sin. N sin. cos. a; and this is the nutation in north polar distance. 244. In finding the nutation in right ascension, n n' may be considered as the measure of the angle subtended at either pole P or P' by the interval between n and m; also sn may be considered as equal to the declination d of the star, and SP' as its complement: then, as in the case of general precession (art. 227.), that is consequently sin. P's P'r :: sin. sn nn', cos. d P'r sin. d: nn'; nn' P'r tan. d. But P'r PP' sin. P'Pr PP' cos. (P'PR + a) therefore PP' (cos. P'PR cos. a sin. P'PR sin. a) =9".239 cos. N cos. a + 18′′.03 sin. N. sin. Ø sin. a; nn' = (9′′.239 cos. N cos. a + 18."03 sin. N. sin. Ø sin. a) tan. d. This value of nn' being added to the above value of Et gives the required lunar nutation in right ascension. 245. The stars are known to suffer an apparent displacement depending on the position of the sun with respect to the point of the vernal equinox, and similar to that which depends on the moon's node; but, on account of its smallness, it is incapable of being detected in a separate form by actual observations, and is only known as a result of theory. Its effects may be represented by a motion of the pole of the world in an ellipse about its place in that which depends on the moon: the major axis of such ellipse is 0".435 and the minor axis is 0.399. But, since the inequality is compensated twice in each year, being the greatest at the times of the solstices (June and December) and vanishing at the times of the equinoxes (March and September) the angle corresponding to APE (art. 241.), instead of being equal to the excess of 360° above the longitude of the moon's node, is equal to twice the sun's longitude. CHAP. XI. THE ORBITS OF PLANETS AND THEIR SATELLITES. VARIABILITY OF THE PROCESSES FOR FINDING THE ELEMENTS OF PLANETARY ORBITS. 246. THE distance of a planet from the sun may be determined approximatively in terms of the earth's distance from that luminary by various processes. For example, if s represent the sun, E the earth, and v one of the inferior planets, Mercury or Venus, supposed to revolve in a circular orbit about s, in the plane of the ecliptic; and if the angle SEV be observed with an instrument at the time when the planet appears to be at its greatest elongation from the sun, or if SEV be taken to represent the difference between the longitude of the sun and the geocentric longitude of the planet at that time; then the visual ray VE being a tangent to the orbit of v, and consequently at right angles to sv, we have (Pl. Trigon., art. 56.) SV SE sin. SEV. Thus sv, the required distance, may be found. E With respect to a superior planet, its distance from the sun, in terms of the distance of the latter from the earth, might be computed approximatively by means of observations made at times when the planet appears to be stationary, before and after its apparent movement is in retrograde order. Thus, let s be the sun, P a superior planet, supposed to be in the plane of the ecliptic; and let E be the place of the earth in its orbit when a tangent to the latter would pass through P: at this time P appears to an observer at E to be stationary in the heavens, the apparent movement being subsequently retrograde or from p towards p'. At the same time let the arc E' P E sp, the apparent distance of P from a fixed star s in the plane of the ecliptic, be found by an angular instrument. Now if, for the present, the motion of P towards P', in its orbit, be neglected; when the earth, having described the arc EAE', arrives at E' where a tangent to its orbit would also pass through P, the planet is again stationary; and in this situation of the earth let the arc sp', or the apparent distance of P from s, be measured. Then EP or SP being very small compared with the distance of s from E or s, the arc pp' (=sp-sp) may be considered as the measure of the angle pPp' or EPE', and half of this arc or angle may be considered as equal to EPS: therefore, in the plane triangle PES, right angled at E, the distance SE being given, the line SP or the distance of the planet P from the sun may be computed. But while the earth is moving from E to E', the planet will have moved from P to P'; in which case the apparent angular distance of P' from s will be sp", and the angular movement of P about the sun, that is the angle PSP', which may be considered as equal to PE'P', must be added to the arc pp" obtained from the observations, in order to have the arc pp', or the angle EPE'. The distance of the planet Mars from the earth may be obtained by means of its diurnal parallax; the latter being deduced from corresponding observations at two points on the earth's surface as described in art. 160., and such observations are made by astronomers when opportunities present themselves. 247. The observed right ascensions and declinations of a planet enable an astronomer to obtain a series of points in the celestial sphere, which being supposed to be joined, there will be formed a trace of the line of the planet's motion in space, as it would appear to an observer on the earth's surface; and neglecting the diurnal parallax, it may be considered as the trace of the geocentric path. But if it be assumed that the planets revolve about the sun as the centre, or a focus of their orbits; then, preparatory to determining the forms of the orbits, the places which a planet seems to occupy in the heavens, when viewed from the earth, must be reduced to the places which it would appear to occupy if seen from the sun: with respect, however, to the longitude of a planet, no such reduction is necessary when the planet is in conjunction with, or in opposition to the sun; the geocentric longitude being then the same as the helio |