In this manner the motion of the nodes of all the satellites upon the orbits of each other, from their mutual attractions, may be determined, granting the quantity of matter in each to be rightly assumed; and then that motion (1077) may be referred to the plane of Jupiter's orbit. 1099. But the nodes have a further motion, arising from the disturbing force of the sun, which may be calculated in the same manner as the motion of the moon's node has been. But knowing the motion of the moon's nodes, the motion of the nodes of the satellites may be more easily found by Art. 854; for if p the periodic time of the moon, P=that of the sun, p'the periodic time of a satellite of Jupiter, and P' the time of the revolution of Jupiter about the sun, M=the motion of the nodes of the moon, and m the cotemporary motion of the nodes of the satellite; then P : P2 × p p p2 P' :: M : m = M × 201 Ex. The mean annual motion of the moon's nodes 19°. 19′ (938); also, p=27,32 days, P = 365,256; for the second satellite of Jupiter p' = 3,55, and P'=4332,6; hence, m=19°. 19′ × 365,2563 × 3,55 = 1'. 4" the mean an 2 4332,6 x 27,32, nual motion of the node of the second satellite of Jupiter upon the plane of Jupiter's orbit. Thus we may find the whole motion of the nodes of all the satellites. 1100. The quantities of matter of the satellites of Saturn not having been determined, we cannot find the motion of their nodes from their mutual attractions; but we can find their motion arising from the disturbing force of the sun, in the same manner as we find the motion of those of Jupiter. M. de la PLACE, in his Théorie du Mouvement et de la Figure Elliptique des Planetes, has given the following theorems for the secular inequalities of the planets. 1101. Let a planet P be acted upon by another planet p, and let m be the quantity of matter in p, that of the sun being represented by unity; via ley mus a -the mean distance of P from the sun; ex a=the excentricity of its orbit; 3 L =the longitude of its aphelion at any given epoch; r =the longitude of its ascending node on a fixed plane at the same epoch; =the inclination of its orbit upon the same plane; ん THE PLANETS, FROM THEIR | MUTUAL ATTRACTIONS. 1n; And let a”, exď, L', and be the same in respect top, and let Jar Ladrig bon919b ed you enouterte lamia usit mon multoɑ dory (~701) portom tedt godt bar, bomuzes vlide radot dono fu 19firm" vhuse, 1 1 2n 1 + n22 3 1 + n2 3n x ( 3 4.3-1 3 4.52-1 X +ገ 3 2n × 104 Hobo oxt to moito &c. 1+22 1102. Let v be the number of revolutions of P from the given epoch, v being negative for any time before the epoch; then m xv x 360° x e'x sin. L'-L xcx 1+n*=3bn. #7sh £2.r£=q The mean motion of the aphelion according to the order of the signs is to scale ed acq q aoqu mxv x 360° x ( nc- 2 x cos. I'-Lxcx1+n'-3bn). tillstez 1199The diminution of the inclination of the orbit upon the fixed plane is to lentum et moit show to moitom sd bait toaso #fresh to 9910 gaidunei 4 mxv× 360° x ncr sin. F'bar, meɔ ow tud tique to out to tom :) but 37 25 15mm Jingz sili 255 The retrograde motion of the node upon the fixed plane is des edt lo esitlupeni retos ont fol moroodt gaiwollot 9dt 90 we gult to tedt 9 mostem to vtip If we thus collect the effects of all the planets upon P, the sum will give the whole variation of the elements of the orbit of P. 2009 evig var to oiladas esi to gbutjenol eft:1103. If we fix upon the year 1700 for the epoch, we shall for the diminution of the obliquity of the ecliptic, produced by the action of P upon the earth; v being the number of revolutions of the earth from 1700, y the inclination of the orbit of the planet p upon the ecliptic, and r' the longitude of the ascending node. The sum of all the diminutions from all the planets will give the whole diminution. When r' is greater than 180°, which it is not at present for any of the planets, the sin. r' becomes negative, and the obliquity will then be increased by that planet. 1104. Hence, by collecting the effects of all the planets, if u be the number of years after 1700, the whole diminution will be the difference between 932",56 and 932",56 x cos. 17",7686v — 3140′′,34 x sin. 32",8412v, supposing the variation at this time to be 50 in 100 years. For any number of years before. 1700, the sign of the second term must be changed. 1105. In like manner, all the planets together will produce a precession of the equinoxes equal to 50",53353v – 3292′′,28 sin. 17′′,7686v-9315′′,65 cos. 32′′,8412v + 9315",65 in v years after 1700; for any time before, the sign of v becomes negative. The inequality of the precession of the equinoxes changes the secular motion of the sun in respect to the equinoxes; this motion is 46′ in this age, but it was 45'. 23" in the beginning of our æra. Hence, the place of the sun, calculated with the uniform secular motion of 46', as in our Tables, will require a secular equation. And this secular motion of the sun gives, for the increase of the year by going back from 1700, 36′′,114 × sin. 32′′,8412v + 6",9039 x cos. 17",7686v-6",9039. Hence, at the time of HIPPARCHUS the year was about 10",33 longer than at this time (1088). CHAP. XXXVII. ON THE EFFECTS PRODUCED ON THE MOTIONS OF THE PLANETS IN THE PLANES Art. 1106. LET S be the center of force, PM the orbit of a body described by virtue of two forces, one (f) tending from the body at P in the direction PS, and the other (F) acting in a direction P perpendicular to PS. Let SQ be a line given in position; draw PA perpendicular to SQ, and complete the parallelogram PASV, and draw Vr, At perpendicular to PS. Of the whole force acting on P, let M be that part which acts in the direction PA, and m the part acting in the direction PV; also, let r= PS, p=PA, q=PV, i=the fluxion of the time, v=the angle PSQ, x=its sine, y=its cosine; then Mr= that part of the force M which acts in the direction PS, and my that part of the force y which acts in the same direction; hence, Mx+my=ƒ; also, My ,,1། that part of the force M which acts in a direction parallel to tЛ, and mx= that part of the force m which acts in a direction parallel to rV; hence, mrMy=F. Now by the principles of motion*, p=— Mi2, and q=—mi, the fluxion of the time being constant. But p=ræ, and q=ry; and as ́i =yv, and y=r, we have, p=xr+ryv, (A) p=xr+2yrv+ryö+ræv2=− Mi ; q=yr-rxv, (B) q=yï— 2xrv—ræö—ryv2 — — mta ; Multiply (4) by y and (B) by x, and subtract the latter from the former, and we get (C) 2rö+rö= −ťa × My — mx — Fi3. If F represent any accelerative force, v the velocity which the body would acquire in 1" by that force, s the fluxion of the space described in a given time with that velocity, then if we measure the velocity by the space described uniformly in 1", we have 1": :::st, therefore ; but varies as Fxi; let therefore v=Fxi, ands = ±F2. FIG. 239. Multiply (A) by ≈ and (B) by y and add them together, and we get From the equations (C) and (D), the curve PM described by the body P may be found. These fluxional equations are the same as those determined by CLAIRAUT, EULER and MAYER, in their Treatises upon the theory of the moon, the integration of which is a problem of great difficulty; and as the first of these Authors has proceeded in a manner the most easy to be understood by the generality of readers, I shall here enter into a very full explanation of all his principles of investigation. 1107. Let the force ƒ consist of two parts, one of which = C C being a constant quantity, and the other =D; or let ƒ==+D; then 210+rö=Fi”; C andrï3 — F = 2+Dxt. 2rri rö t C + D x 1. Multiply the first of these equations by, and we get – Fri, whose fluent (¿ being constant) is "=a+ƒFri, a being a constant quantity. Multiply this equation by Fri, and we have Fr3¿=aFri+ Frix/Frt, whose fluent is ƒFriva ƒ Fri + }ƒ Fri* •; multiply this by 2, and add a to both sides, and it becomes a +2 Fr3¿ = a + 2a ƒ Fri +f Fri, whose square root is ✔✅a+2ƒ Fr3v=a+f Fri. Hence, consequently i= r2¿ = (if we put e= Frii] a* C 1108. Let us next take the other equation rỡ –ï= +Dxt. Now in this equation, i is constant and variable; but it is well known, that in a fluxional equation of the second order, where r is variable and constant, that if we substitute for ", the equation will be changed into one in which is constant and i variable; if therefore for we substitute ï—””, the equation will be changed into one in which rand i are both variable. Substituting there * For if y=ƒ Fri, then y = Fri, therefore yy = Fri×f Fri, and the fluent=} y2=}ƒFri”‚...‚. |
