Now equations (2) show that the latus rectum of the in 2m2 dm stantaneous parabola is ; and as is negative, by (4), dt These give for the co-ordinates of the focus of the instantaneous orbit da dt' If these expressions be differentiated and &c. be eliminated by means of (4) and (5), it will be seen that da dt . is negative, or the axis of the instantaneous orbit moves backwards, until the particle reaches the vertex; after which dy' it progresses for the rest of the motion; also that is dt positive if m>n, that is, the focus of the instantaneous orbit moves upwards while the direction of motion of the particle makes with the horizon an angle less than 45°, i. e. while the particle is above the latus rectum of the instantaneous orbit. 300. A particle, moving about a center of force whose intensity is inversely as the square of the distance, is subjected to a small disturbing force in its plane of motion; to investigate the change in the form and position of the orbit. Let the disturbing force be resolved into two, & and y, one along the radius vector and the other perpendicular to it; the equations of motion are do 2 dt 1 d (rede) = ↓ dt .(1). if we omit the forces and . When we consider their effect then, the quantities h, e and must be considered variable. But in the instantaneous orbit, the velocity and direction of motion are the same as in the actual orbit, and therefore if (2) be differentiated, considering h, e, and a variable, the dr de results for r, and must be the same in form as if the do' dt disturbing forces had not acted. This will enable us to avoid second differential coefficients of h, e and ; and the substidr dr tution of their values for dt' dt altogether three equations for ᏧᎾ and in (1), will give us dh de da dt' dt' dt dt The expressions for these quantities are complicated and so we do not give them. They will be more easily investigated in particular cases, when & and are given. == In the case of the orbit being an ellipse, h2= ua (1 − e2), so that we have by substitution And the second integral of the second of equations (1) involves e or the epoch, which will also be thus found as a function of t. 301. If we desire the change produced in the form and position of an orbit by a slight change made in the velocity, or direction of motion, &c. at some particular point, we must express separately each of the elements of the orbit in terms of the quantity to be changed; then taking the differentials of both sides, we have the required changes of value. T. D. 21 Thus, we have generally in an elliptic orbit v2 = μ (2-1). § 142 (9). At the extremity of the axis major farthest from the focus this becomes Now if at this point V be made V+ SV, without change of direction, we have the condition that in the new orbit a (1+e) shall have the same value as in the old; since this will still be the apsidal distance. which determine the increase of the axis major and diminution of the excentricity, and the same method is applicable to more complicated cases. Again, in the case of a parabolic orbit, as in Chap. IV., it is easy to see that a change in the magnitude of the velocity shifts the focus in the line joining it with the projectile through a space VSV raises the directrix through an equal space, and increases the latus-rectum by where a is the inclination of the path to the horizon at the instant of the impact. If the direction of motion only be changed, the directrix is unaltered, the focus moves in a direction perpendicular to the line joining it with the projectile, and the latus rectum is diminished by the quantity 4V2 In the latter case the new orbit again intersects the old, and the tangents to either at the two points of intersection are at right angles to each other; so long as the displacement da is indefinitely small. These results may easily be extended by geometrical processes, as in Chap. IV., or deduced by differentiation from the analytical results there given. EXAMPLES. (1) If a small velocity n be impressed on a planet, in με h the direction of the radius vector, shew that Sene sin (0), δι =-n cos (0-w). (2) A satellite moves about a spherical planet in the plane of its equator, in a slightly elliptic orbit. Find the motion of the apse due to an uniform mountain ridge at the equator. (3) Central force varying as the distance, the velocity of a particle is increased by th when it is at the extremity of n one of the equal conjugate diameters of its orbit. Shew that (4) At what point of an elliptic orbit described about the focus, can a small change be made in the direction of motion without altering the position of the apse? If so be this change, shew that (in the supposed case) (5) Shew that in an elliptic orbit about the focus, if the as the particle is moving to or from the nearer apse. (6) A particle moving about a center of force in the focus, in an ellipse of small excentricity, receives a small impulse perpendicular to its direction of motion at any instant. Find the effect on the position of the apse. (7) Again, if at the extremity of the axis minor the velocity be increased by th, and the direction changed so 1 that h remains the same, find the alteration in the form and position of the orbit 108 |