ASTRONOMY.

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Earth as it proceeds from Mtor, its path will be within the circle,but the solar influence from r to B and t, tends to keep the Moon further from theEarfh, which, together with the increase of velocityfrom M to B, will cause the Moon to proceed beyond the circle to-wards C, and as the same causes operate in the other half of theorbit, the Moon will describe an elliptical curve, being furthestfrom the Earth in the quadratures and nearest in the syzygies.—The effects here described are such as would take place on thesupposition of the Moon ’s orbit being at first a circle in the planeof the ecliptic; but because the orbit is an ellipse, and inclined to theplane of the ecliptic, some further irregularities will takeplace. Letthe Moon’s orbit be an ellipse, and let E (Fig. 4,) represent the Earth in one of the foci; then because the disturbing force from E to Gbrings the Moon nearer the Earth , it will come to the apsis before itarrives at M ; as suppose at o, hence the line of the apsides will

f o back from M to A ; but because the Sun’s action lessens thelarlh’s attractive force from G to 11 more than it increased itfrom Fto G, the apsis will be carried forward as from N to D,more than'it had been carried backward: hence the line of theapsides will move on (lie whole forward, or in the order of thesigns ; they are found to make a revolution in about nine years.

(110.) The Moon ’s nodes, and the inclinalior. of the orbit, willlikewise be subject to variations from the same cause ; for part ofthe disturbing force will be employed to bring the Moon fromthe plane of its orbit to the plane of the ecliptic, and conse-quently to change the position of the nodes of the two forcesm q, m r, (Fig. 1,) into which the disturbing force in o was resolved:mq being in the plane, of the lunar orbit has no influence to bringthe'Moon out of that plane; but on account of the inclination otthe orbits mr will not always be in the plane of the Moon ’s orbit,that is, it will for me greats t part make an angle w ith the tangentof the orbit at m, and the effect of m r on the nodes and inclina-tion depends on the angle contained by m r, m t, or the inclinationof the plane cf m o, with the plane of the Moon ’s orbit.

(111.) In order to form a clear idea of the action of this force,conceive the plane m s r t to pass through m at right angles toEra, it is evident tliL plane will pass through the tangent mt,and tire line m r ; in r is also in the pane r in q o cf, which inter-sects the plane of the Earth’s orbit in tire line ES, and the Moon ’sorbit in E g; hence the inclination of the plane of the Moon ’s or-bit to the plane/o is the angle rmt\ if in the plane if a perpen-dicular be drawn to Mi f, the force m r will be resolved into two,mt and rt, or ms, of which m t being in tire plane of the lunar or-bit, has no other influence than to accelerate or retard the Moon ’smotion ; but sm being perpendicular to that plane tends to bringthe Moon towards, or to remove it from the plane of the ecliptic.It is evident that a tangent to the Moon ’s orbit, as m t, cannot in-tersect the plane of the ecliptic except in the line of the nodes, nei-ther can in r intersect it except in some part of the line S E, sinceS E is the common section of the plane of the ecliptic, and theplane in which m r is situated. When the Moon is in the quadra-tures, m r, and consequently ms vanishes, hence the nodes and in-clination in this case are stationary. When the nodes are in thesyzygies, that is, when the line of the nodes coincides with S E,the plane J o will coincide with the plane of the Moon ’s orbit,lienee m r coincides with the tangent in t, and r t, or m s, vanishes ;hence also' in this case the nodes and inclination are stationary, atall other times they are variable. The nearer the line of thenodes is to the syzygies, every thing else being the same, the lessis the angle r mt, and therefore m t is greater, and tr less ; hencethe force which affects the inclination is greater as the nodes ap-proach the quadrature, where it is a maximum.

(112.) Suppose the line oi the nodes in the situation O A, andthe Moon at in, moving from the quadrature to the conjunction,(Fig. 5.) Then if B be the ascending node the Moon is below theecliptic, and since mt cuts the ecliptic in n, and mr cuts it in p,mr is below mt, therefore the force rt tends to bring the Moon further below the ecliptic, hence it will not intersect the ecliptictill it has past B, that is, the nodes move forward. While the Moon is moving above the ecliptic from/, the node B to tiie quadratureN m t is still above in r, tor from B to P the tangent m t rises aboveE S from some point in E A ; from P to in' in' r' meets the eclip-tic sooner than ni i' till B E m' is a right angle, after which in' Vmeets the ecliptic on the contrary side', and rises entirely above P.Hence from B to N the force t r tends to bring the Moon nearerthe ecliptic by which it intersects it sooner, that is, the nodesVOL 1. —NO. 19.

move backward from O and B towards N and M. If we considerB as the descending node, it may be shewn in the same mannerthat from M to B the force friends to bring the Moon fuitherfrom the ecliptic, and from B to N to bring it nearer the ecliptic.Thus also it may be shewn, that the nodes go forward while theMoon moves from X to O, and backward while it moves fromO to M. If the nodes were in the situation C D, it will appearby like reasoning, that while tire Moon moves from M to D, andfrom N to C, the nodes go backward, but they go forward whilethe Moon proceeds from D to N, and from C'to M. Hence wehave this general rule, viz. while the Moon advances from thequadrature to the next node, the nodes move towards the nearestsyzygies, and while the Moon advances from the node to the nextquadrature, the nodes remove further from the nearest syzygies;and since in every revolution of the Moon , whatever be t'he’situa-tion of the nodes, they move further backward than forward, theywill on the whole have a retrograde motion. They are found tomake a revolution in about 19 years.

(113.) It is evident from the above that the inclination of theMoon ’s orbit is almost continually changing : let the nodes be inthe second and fourth quarters as at O and B, (Fig. 6,) since, asabove explained, while the Moon move? from B to N it is broughtnearer to the ecliptic by tire force i r, which affects the nodes andinclination ; hence in going through the quadrant BD, it will bekept nearer the ecliptic, that is, it will depart from the eclipticmore slowly, or its inclination will decrease ; from I) to N it willapproach the ecliptic more quickly, or its inclination will increase.Again from N to O the force tr tends to bring the Moon fuitherIroni the ecliptic, which it therefore approaches more slowly, orthe inclination decreases ; now the arc NO together with the qua-drant BD is always greater than DN, therefore the indination willon the whole decrease: for the same reason while the Moon movesthrough the quadrant OC and the arc MB the inclination de-creases, and increases in tiie arc CM : hence while the line of thenodes is in the second and fourth quarters the inclination de-creases, and is least at the quadratures. By the same kind of argu-ment it will be found that when the nodes are in the first and thirdquarters, as at C and D, the inclination in the arc CM and qua-drant BD is increased and decreased in MB, and the same willapply in the half DOC; therefore while the nodes recede from thequadratures to the syzygies the inclination is on the whole in-creased, as before it was decreased, and is greatest when the nodesare in conjunction.

(! 14.) The line of the conjunction is by the Earth ’s annual mo-tion brought into every possible situation with respect to the nodesin one year, independent of their own proper motion ; which lastoccasions the change of situation to be performed in about nine-teen days less ; this will be illustrated by an inspection of the 7thfigure. Let the Earth be at C, and the line of the nodes VX inconjunction with the Sun Z; while the Earth moves from C to Ii,the nodes are in the second and fourth quarters ; hence, as above .shewn, the inclination decreases, and is least at E, where the nodesare in quadrature; from E to G it increases, because the nodes, arein the first and third quarters, and thus from G the inclination les-sens till the Earth comes to A, and increases during the next threemonths, when the Earth has again arrived at C. This supposes thatthe nodes on the whole are stationary, but (as before observed) thenodes make a retrograde revolution in about nineteen years; hence,when the Earth is at K, about nineteen days before the end of theyear, the nodes are in conjunction, and all these pfuenoinena arecompleted ; from K the inclination begins again to diminish.

(115.) When the Earth having performed a revolution has cometo C, the nodes will be in the line L)1, which has moved back-ward 19 “ from the position XV, which it had when the Earth setset out from C ; in another year, or revolution of the Earth , it willcome to the position NO, so that in five years it will have pastthe quadratures at C, in ten years the other node Y will be alittle past X, so that in nineteen years it will come again nearly tothe same situation XY r , the inclination as is evident, having twiceperformed its variation of increase and diminution, the Earth beingat C. All these irregularities will he greater as the distance ofthe Bun is less in proportion to the distance of the Moon : hem-ethey are somewhat greater in winter than in summer, an.d also.when the Moon is between the quadratures and conjunction, thanwhen nearer the opposition. The same reasoning Will apply tothe other secondary planets, but the disturbing force will be less