Thus if the attraction vary directly as the nth power of the distance, we have This shews that n cannot be less than 3, or that the attraction must vary according to a lower inverse power of the distance than the third, if the circle with the centre of attraction at its centre is to be an approximation to the path of the particle: and the investigation furnishes a simple example of the determination of the conditions of Kinetic Stability, which we cannot discuss in this elementary treatise. To find the law of attraction that the apsidal angle in the nearly circular orbit, whatever be its radius, may be equal to a given angle, a suppose, we have 3 and therefore the law of attraction, μu3 þ (u), is μu3 ̄ã3. Thus for aπ we have the law of the inverse square of π the distance, for a= the law of the direct distance, while π 2 a= corresponds to a constant central attraction. √3 If 1 co'c фе be zero or negative, the form of the integral of (b) above shows that a does not remain infinitely small; i.e. that a circle is not a kinetically stable path under the conditions. In this case all that (b) can furnish is an account of the way in which the orbit begins to differ from a circle in consequence of a slight disturbance. 149. A particle is projected from a given point in a given direction and with a given velocity, and moves under the action of a central attraction varying inversely as the square of the distance; to determine the orbit. We have Puu, and therefore = This is the polar equation of a conic section, the focus (the centre of force) being the pole. Let R be the distance of the point of projection from the centre; ẞ the angle, and V the velocity, of projection; then when 0 = 0, Now (1) is the general polar equation of a conic section focus the pole; and, as its nature depends on the value of the excentricity e given by (4'), we see that 150. By § 102, the square of the velocity from infinity at C distance R, for the law of attraction we are considering, is 2μ R and the above conditions may therefore be expressed more concisely by saying that the orbit will be a hyperbola, a parabola, or an ellipse, according as the velocity of projection is greater than, equal to, or less than, the velocity from infinity. Illustrations of this proposition are found in the cases of comets and meteor swarms. The velocity of a particle moving in a circle is also often taken as the standard of comparison for estimating the velocities of bodies in their orbits. For the gravitation law of attraction the square of the velocity in a circle of radius R μ is ; and the above conditions may be expressed in another R T. D. 9 form by saying that the orbit will be a hyperbola, a parabola, or an ellipse, according as the velocity of projection is greater than, equal to, or less than, √2 times the velocity in a circle at the same distance. 151. Supposing the orbit to be an ellipse, we shall obtain its major axis and latus rectum most easily by a different process of integrating the differential equation. Multiplying it du and integrating, we obtain by h2 de Now to determine the apsidal distances, we must put which is a quadratic equation whose roots are the reciprocals of the two apsidal distances. But if a be the semi-axis major, and e the excentricity, these distances are a (1-e) and a (1 + e). Hence, as the coefficient of the second term of (6) is the sum of the roots with their signs changed, we have or or And, as the third term is the product of the roots, and therefore 2 Ꭱ Equations (7) and (8) give the latus rectum and major axis of the orbit, and shew that the major axis is independent of the direction of projection. Equation (9) gives a useful expression for the velocity at any point, and shews that the radius of the circle of zero velocity is 2a. 152. The time of describing any given angle is to be obtained from the formula, From this, combined with the polar equation of a conic section about the focus, we have |