182 CHAPTER IV. ON THE ELLIPTICAL MOTION OF THE PLANETS. 359. THE elliptical orbit of the earth is the plane of the ecliptic: the plane of the terrestial equator cuts the plane of the ecliptic in a line passing through the vernal and autumnal equinoxes. The vernal equinox is assumed as an origin from whence the angular distances of the heavenly bodies are estimated. Astronomers designate that point by the character, the first point of Aries, although these points have not coincided for 2230 years, on account of the precession or retrograde motion of the equinoxes. 360. Angular distance from the vernal equinox, or first point of Aries, estimated on the plane of the ecliptic, is longitude, which is reckoned from west to east, the direction in which the bodies of the solar system revolve round the sun. For example, let EnBN, fig. 72, represent the ecliptic, S the sun, and the first point of Aries, or vernal equinox. If the earth be in E, its longitude is the angle SE. 361. The earth alone moves in the plane of the ecliptic, the orbits of the other bodies of the system are inclined to it at small angles; so that the planets, in their revolutions, are sometimes seen above that plane, and sometimes below it. The angular distance of a planet above or below the plane of the ecliptic, is its latitude; when the planet is above that plane, it is said to have north latitude, and when below it, south latitude. Latitude is reckoned from zero to 180°. 362. Let EnBN represent the plane of the ecliptic, and let m be a planet moving round the sun S in the direction mPn, the orbit. being inclined to the ecliptic at the angle PNE; the part of the orbit NPn is supposed to be above the plane of the ecliptic, and NAn below it. The line NSn, which is the intersection of the plane of the orbit with the plane of the ecliptic, is the line of nodes; it always passes through the centre of the sun. When the planet is in N, it is in its ascending node; when in n, it is in its descending node. Let mp be a perpendicular from m on the plane of the ecliptic, Sp is the projection of the radius vector Sm, and is the curtate distance of the planet from the sun. ∞ SN is the longitude of the ascending node; and it is clear that the longitude of n, the descending node, is 180° greater. The longitude of m is Sm, or Sp, according as it is estimated on the orbit, or on the ecliptic; and mSp, the angular height of m above the plane of the ecliptic, is its latitude. As the position of the first point of Aries is known, it is evident that the place of a planet m in its orbit is found, when the angles Sm, mSp, and Sm, its distance from the sun, are known at any given time, or Sp, pSm, and Sp, which are more generally employed. But in order to ascertain the real place of a body, it is also requisite to know the nature of the orbit in which it moves, and the position of the orbit in space. This depends on six constant quantities, CS AP, the greater axis of the ellipse; the eccentricity; CP' SP, the longitude of P, the perihelion; SN, the longitude of N, the ascending node; ENP, the inclination of the orbit on the plane of the ecliptic; and on the longitude of the epoch, or position of the body at the origin of the time. B These six quantities, called the elements of the orbit, are determined by observation; therefore the object of analysis is to form equations between the longitude, latitude, and distance from the sun, in values of the time; and from them to compute tables which will give values of these three quantities, corresponding to any assumed time, for a planet or satellite; so that the situation of every body in the system may be ascertained by inspection alone, for any time past, present, or future. 363. The motion of the earth differs from that of any other planet, only in having no latitude, since it moves in the plane of the Α fig. 72. n m In conseecliptic, which passes through the centre of the sun. quence of the mutual attraction of the celestial bodies, the position of the ecliptic is variable to a very minute extent; but as the variation is known, its position can be ascertained. 364. The motions of the celestial bodies, and the positions of their orbits, will be referred to the known position of this plane at some assumed epoch, say 1750, unless the contrary be expressly mentioned. It will therefore be assumed to be the plane of the co-ordinates x and y, and will be called the Fixed Plane, Motion of one Body. 365. If the undisturbed elliptical motion of one body round the sun be considered, the equations in article 146 become + = 0, where μ is put for Sm, the sum of the masses of the sun and planet, and r = √ x2 + y2 + zo. In these three equations, the force is inversely as the square of the distance; they ought therefore to give all the circumstances of elliptical motion. Their finite values will give x, y, z, in values of the time, which may be assumed at pleasure: thus the place of the body in its elliptical orbit will be known at any instant; and as the equations are of the second order, six arbitrary constant quantities will be introduced by their integration, which determine the six elements of the orbit. 0= = 0, 366. These give the motion of the planet with regard to the sun; but the equations d2x 0= dt d2y = 0, my = d2z (89) mz dt2 of article 346, give values of x, y, z, in terms of the time which will determine the motion of the sun in space; for if the first of them be multiplied by S + m, and added to consequently, + a = a + bt (S+m) r 2.3 y = a + b't xd'y xdy z=a"b"t + m dt These equations give the motion of the sun in space accompanied by m; and as they are the same for each body, if Em be substituted for m, they will determine the absolute motion of the sun attended by the whole system, when the relative motions of m, m', m', &c., are known. mx S+ m 367. But in order to ascertain the values of x, y, z, the equations (89) must be integrated. Since these equations are linear and of the second order, their integrals must contain six constant quantities. They are also symmetrical and so connected, that any one of the variable quantities x, y, z, depends on the other two. M. Pontécoulant has determined these integrals with great elegance and simplicity in the following manner. 368. If the first of the equations (89) of elliptical motion multiplied by y, be subtracted from the second multiplied by x, the result will be yd x = 0, ydx d2x dt2 my S+ m "" ; mz S+ m = C. = 0; = 0, In the same way it is easy to find that zdr xdz = c'; a If where c, d, c', are arbitrary constant quantities introduced by integration. Again, if the first of the same equations be multiplied by 2dr, the second by 2dy, and the third by 2dz, their sum will be 2dxd'x+2dydy+2dzdz 2ụ (cdr+?dy+d:) + = 0. dt p3 But r2 = x2 + y2 + z2 ; whence <= 0, " a being an arbitrary constant quantity. day dt - dt = = C= c'd2x-c'd y ydz - zdy multiplied by c' = , multiplied by c' = μ (rdz — zdr) μ. and by a similar process values of (xdz – zdr) – Hy (zdy — ydz) = = μα c'dx-c"dy = d', ydz -zdy d, and ud, r T r may be found, the integrals of which are d= ; c= dt (90) |