is very small, and consequently that the square of the term Tdv compounded of it, may, in the expansion of the denominator of II, Now, since u is, by hypothesis, constant, the sole thing that remains to be done, in determining the value of П, is to find the values of cos. 2 w and off sin. 2w.dv: and this can be done, if we can express 2 w in terms of v. Values of cos. 2 w, and sin. 2 w. The angle w, or LTS, is the difference of v and v. Now u} and, when the orbits are circular, are proportional to the mean Ꮮ motions of L and T': let those mean motions be In the above value of II, the only variable term is a cosine of a multiple of the arc v: therefore, (see p. 104.) the differential equation can be integrated, and which is the first approximate value of u, and shews, that the effect of the disturbing force is to render u variable, or, in other words, to destroy the circular form of the orbit. We may give a different form to the preceding expression by substituting for the values of h2, and of m' a3 Since, by hypothesis, the disturbing force Tis very small, the two expressions for the element of the time (see pp. 95, 96.) in the elliptical and disturbed systems will be nearly equal: the value of h, therefore, will be nearly the same in both. But, in the former, that value (see p. 25.) is equal a.(1-e), a representing the mean distance. If the system, however, be disturbed by the action of a third body, the constant distance which corresponds to a will be changed: consequently, if we choose to retain the symbol a still to denote this latter distance, we must express the former mean distance by some other symbol: suppose it to be a. Hence we have, in the case of three bodies, With regard to the second point, the value of to p. 109, we shall find that and the value of this, when the body is in quadratures (when mean force of the gravity of the Moon to the Earth, or, the centri force in its mean value; this will be hereafter denoted by the symbol K. In specific cases the arithmetical value of the disturbing force, in its mean value, is easily expounded: for instance, in the case of the Moon, m' denoting the mass of the Sun. The force by which the Earth is drawn towards the Sun = m But this same force, by Newton, Prop. 4. and p. 29. = and similarly, when m denotes the Moon's mass, and M the Earth's, or, if expressed by a vulgar fraction, the mean value of the disturbing force, when the mean gravitation of the Moon to the Earth is represented by 1. If we now substitute the preceding values for h2 and the expression for u, (see p. 130.), we shall have Astronomy, pp. 305, 306. See Newton, Lib. III. Prop. xxv. supposing a = a, and u to = - when v = 0. nearly, This is the first approximate value of u: and it shews us that the former constant radius is, by the action of the disturbing force, rendered variable. α. is the radius of the circular orbit in which, when the disturbing force is excluded, the body is supposed to revolve: and a is the constant part of the radius vector in the disturbed orbit, and between a and a, when v = 0, we have this equation, If, from the above formula, we wish to compute the Moon's radius (supposing that to be the body disturbed) in conjunction, opposition, and quadratures *, we must substitute respectively for v, 0, 180o and 90o. The equation to an ellipse, (see p. 27.) is of this form |