dy, dz, 82,). da But if (dx,) + dx,, (dy,) + dy,, (dz,) + dz,, be put, in equations (87), in place of their equals, (105) Since (dx), (dy,), (dz,), are supposed to satisfy these equations when R = 0. If the preceding values of dx,, dy,, dz,, be put in equation (105), it becomes identical with equation (104). Hence the integral (103) satisfies the equations (87), whether the disturbing forces be included or not, the only difference being that, in the first case, a must be regarded as a variable quantity, and in the last it is constant. The same may be shown of all the first integrals of equations (87), when R is zero. 418. It appears, from what has been said, 1st, that as the motion is performed in the unvaried ellipse during the first element of time, x, y, z, dx, dy, dz, are alike in the varied and unvaried ellipse. 2nd, That as the motion is performed in the variable ellipse during the second element of time, if dx, d'y, dz, be considered as belonging to the unvaried ellipse, d'r + dồr, d3y + dồy, dz + dồz will belong to the variable orbit of m. Hence the differential equation of the first order, which determines the motion of the body, answers for both orbits during the first instant of the time, the elements of the orbit being constant; in the second increment of time, the equations of elliptical motion have the form d'v the elements of the orbit being constant; but in the troubled orbit they have the form d2v + n2v + R = 0, dt2 where the elements of the orbit are variable, and R is the part containing the disturbing forces. 419. As the elements of the orbit only vary during the second increment of the time, their variation is of the first order; that is, the eccentricity e becomes e + de, the inclination becomes + dp, &c. &c. 420. The elegant theory of the variation of the arbitrary constant quantities is due to Euler. La Grange first applied it to the celestial motions. 421. It is proposed, first, to determine the periodic and secular variations of the elements of orbits of any eccentricities and inclinations; in the second place, to find those of the planets and satellites, all of which have nearly circular orbits, slightly inclined to the plane of the ecliptic; and then to determine the periodic inequalities, dv, Ss, dr, in longitude, latitude, and distance. Variation of the Elements, whatever the Eccentricities and Inclinations may be. 422. All the elements of the orbit have been determined from the seven arbitrary constant quantities, c, c', c'', f, f', f", and a, introduced by the integration of the equations (87) of elliptical motion ; but it was shown that the elements of the orbit, as well as the differentials dr, dy, dz, vary during the second element of time by the action of the disturbing forces, and then the differentials of the equations (91) will afford the means of finding the variations of the elements, whatever the eccentricities and inclinations of the orbits may be. Equations (87) give which are the changes in dr, dy, dz, due to the disturbing forces alone, the elliptical part being omitted. If, therefore, the differentials of equations (91) be taken, considering c, c', c'', f, f', f", a, dx, dy, dz, alone as variable, when the preceding values of d2x, d'y, dez are substituted, they become 423. If values of c, c', c'', f, f', f", derived from these equations, be substituted instead of their constant values in equations given in article 374 and those following, they will determine the elements of the disturbed orbit. The equations d'x + c'y + cz = 0, μr - h2 +f"x+f'y + fz = 0; με and their differentials c''dx + c'dy + cdz = 0, μdr+f"dx +f'dy + fdz = 0, will also answer in the disturbed orbit, provided the same substitution be made. 424. The mean distance a gives the mean motion of m, or more correctly that in the disturbed orbit, which corresponds with the mean motion in the elliptical orbit; for If be the mean motion of m, then in the elliptical orbit, dendt; but this equation also answers for the disturbed orbit, since the two orbits coincide during the first instant of time. ddy = dndt, But 425. The seven arbitrary constant quantities are only equivalent to five in consequence of the two equations These also exist in the disturbed orbit, when the arbitrary quantities are replaced by their variable values. 426. Since R is given in article 347, all the elements of the disturbed orbit are determined with the exception of e, the longitude of the planet at the epoch. From the equations |
