Hence, the attraction of the whole sphere is precisely the same as if the whole mass were condensed into its centre. COR. 1. The attraction of a shell radius = c and thickness Sc will be obtained from the preceding expression by differentiating it with respect to c, and is attraction of shell = Απρε. δε mass of shell a2 COR. 2. Therefore, the attraction of a heterogeneous sphere on an external particle will be the same as if the whole mass were condensed into its centre, provided the density be the same at all points equally distant from the centre, for then the whole sphere may be considered as the aggregate of an infinite number of uniform shells, and by Cor. 1, each acts as if condensed into its centre. 7. Let us now consider the case of one sphere attracting another. Suppose P in the preceding article to be an elementary particle of a sphere M', whose centre O' suppose at a distance a from O. Then, since action and reaction are equal and opposite, P will attract the whole sphere M just as it would do a particle of mass M placed at 0; the same is true of all the elementary particles which compose the sphere M', therefore the sphere M' will attract the sphere M as if the whole mass of the latter were condensed into its centre : but the attraction of the sphere M' on a particle O is the same as if the attracting sphere were condensed into its centre O'; therefore, Two spheres attract one another as if the whole matter of each sphere were collected at its centre. 8. This remarkable result, which, as may be shewn, holds only when the law of attraction is that of the inverse square of the distance, or that of the direct distance, or a com bination of these by addition or subtraction, reduces the problem of the sun, earth, and moon to that of three particles; the slight error due to the bodies not being perfect spheres will here be neglected, being of an order higher than that to which we intend to carry the present investigation: the error however, though very small, is appreciable, and if a nearer approximation were required, it would be necessary to have regard to this circumstance. (See Appendix, Art. 100.) (9) CHAPTER II. MOTION RELATIVE TO THE EARTH. 9. When a number of particles are in motion under their mutual attractions or other forces, and the motion relatively to one of them is required, we must bring that one to rest, and then keep it at rest without altering the relative motions of the others with respect to it. Now, firstly, the chosen particle will be brought to rest by giving it at any instant a velocity equal and opposite to that which it has at that instant; secondly, it will be kept at rest by applying to it accelerating forces equal and opposite to those which act upon it. Therefore give the same velocity and apply the same accelerating forces to all the bodies of the system, and the absolute motions about the chosen body, which is now at rest, will be the same as their relative motions previously. Problem of Two Bodies. 10. As the sun disturbs the moon's motion with respect to the earth, it is important to know what that motion would have been if no disturbance had existed, or generally :— Two bodies attracting one another with forces varying directly as the mass and inversely as the square of the distance, to determine the orbit of one relatively to the other. Let M, M' be the masses of the bodies, r the distance between them at any time t, M' being the body whose motion relatively to M is required. M The accelerating force of M on M' equals acting towards M' M, while that of M' on M equals in the opposite direction. Therefore, by the principle above stated, we must apply to both M and M' accelerating forces equal and opposite to this latter force, and M' will move about M fixed, the accelerating = twice the area described in a unit of time, e and a being constants to be determined by the circumstances of the motion at any given time. This is the equation to a conic section referred to its focus, the eccentricity being e, the semi-latus rectum μ and the angle made by the apse line with the prime radius a. In the relative motion of the moon, or in that of the sun about the earth, the orbit would, as observation informs us, be an ellipse with small eccentricity, that of the moon being about and that of the sun. 11. The angle 0-a between the radius vector and the apse line is called the true anomaly. If n is the angular velocity of a radius vector which moving uniformly would accomplish its revolution in the same time as the true one, both passing through the apse at the same instant, then nt+-a is called the mean anomaly, e being a constant depending on the instant when the body is at the apse, its value being also equal to the angle between the prime radius and the uniformly revolving one when t=0. Thus, if MT be the fixed line or prime radius, at time t, M' the moving body Mu the uniformly re volving radius at same time, the direction of motion being represented by the arrow. Let MD be the position of Mu when t=0, then M TMD and is called the epoch,* YMA=α= longitude of the apse; therefore, mean anomaly = AMμ=nt +ɛ− a, true anomaly = AMM'=MM'-MA=0-α. 12. To express the mean anomaly in terms of the true in a series ascending according to the powers of e, as far as e*. * The introduction of the epoch is avoided in the Lunar theory by a particular assumption (Art. 34); but in the Planetary it forms one of the elements of the orbit. |