it differs therefore from an ellipse, on account of the term L. cos. (2v - 2 m v). The effect of the disturbing force, then, inasmuch as we are able to infer from the preceding deduced value of u, is not to change the circular orbit into an elliptical. But the inference from that deduced value of u may not, with reference to the real change in the form of the orbit, be strictly true, since that value is only the first approximate one. To ascertain, therefore, the justness of the inference, we ought to deduce a second value of u by substituting the one just obtained in II, and by again integrating the differential equation. But it is easy to perceive, without going through this process, that its result will be, not to rescind terms like L cos. (2v- 2mv), but to augment the value of u by new terms involving the cosines of new arcs: so that, the second equation determining u will be still more remote from an equation to an ellipse than the first is. If we were, however, to substitute the first value of u in II, the second resulting value would contain an arc of a circle without the sign, and be faulty: for, (see p. 95.) one term and since (see p. 129.) the middle term of P is in II is m' u3 2u P h2 u2 when expanded, must contain a term such as N cos. v; there fore II must, and if II contains such a term, then (see pp. 107, &c.), of integration will necessarily introduce, into the value the process ing to the rules laid down in pp. 110, &c. not directly pursue the plain method of approximation, but deviate from it in order to avoid its difficulties. The conditions of the preceding case have been assumed the most simple possible, in order to procure an easy introduction to the solution of the differential equation. The solution that has been obtained gives us merely an imperfect value of u, which, since it represents the inverse of the radius, is proportional (see Astronomy, pp. 95, &c.) to the parallax. The deduction of the value of this quantity is one use then, of the preceding integration: but it is not the chief use: that is to be found in the means afforded us of deducing the longitude (v), which depends, in the first instance, on nt the mean longitude: but t, in order to be determined, requires that u should previously be known, since (see p. 95.) the integral of which cannot be correctly* found, as it is plain, except we know u. The particular method of forming the several terms that represent the true longitude, will be explained in a future part of this Treatise. The present concern is with the differential equation, on which the value of u depends. We shall endeavour to obtain that value by a series of successive corrections. The value of the inverse of the radius of the orbit, from being constant, becomes, by the agency of the disturbing force, (and by the process of one approximation and integration) of this form: and this, as it has been observed, is not the equation to an ellipse: it would be, were the last term rescinded. That last term expounds what in Astronomical language is called an Inequality: the argument is the arc 2 v - 2 m v: and the coefficient is * By this term it is not meant to be understood that, if a correct value of u should be obtained, the integral of the differential can be expressed by a definite equation. It can only be expressed by a series: and, by a reversion of that series, v must be expressed in terms of t. The term E cos. v (see Astron. pp. 322, 324.) expounds what is called the Elliptic Inequality. The rule therefore, for finding u may be expressed either by saying that we must correct its elliptic value by means of the equation represented by L cos. (2 v−2 mv): or, that we must correct its constant value by means of two equations, one due to the elliptic inequality, the other to that inequality of which the argument is 2 v 2 m v. The instance that has been given in this Chapter is one of the most simple that can be imagined when no essential condition is excluded. It will not exactly suit any case in nature: not even that of Venus disturbed by the action of Saturn: still less that of the Moon disturbed by the Sun. It must fail to represent this latter case for several reasons, of which the most prominent are, the eccentricities and inclinations of the Solar and Lunar orbits. Still, on the preceding solution, as on a basis, may more correct ones be founded: by introducing new conditions and by applying corrections proportional to them to the values of P, T, &c. But the method will be best understood by the Example of the succeeding Chapter. • See Astronomy, pp. 324, &c. CHAP. IX. Continuation of the Solution of the Problem of the Three Bodies: the Orbit of the disturbed Body is supposed to be Elliptical: the resulting Value of the Radius Vector thereby augmented with additional Terms. Clairaut's First Method of determining the Progression of the Lunar. Apogee. EXAMPLE 2. IT is required to find the inverse of the radius vector (= -), when the body, revolving in an elliptical orbit of very small eccentricity, is disturbed by the action of a very remote body which revolves in a circular orbit, the plane of which is coincident with that of the elliptical orbit. Into this Example, only one (see Ex. 1. p. 128.) new condition is introduced, namely, the eccentricity of the disturbed orbit, and that is supposed to be very small. neglecting the product of the first and last terms of the two factors. S forces) the square of the disturbing force: if we neglect them, by reason of their minuteness, and retain solely the first term, then This value of II differs from the former, (see p. 128.) by the first term, which is here retained; since, when is not constant du u but the radius vector of an ellipse, is not equal 0. The other dv condition, that of the smallness of the disturbing force, is the same in both cases, and on that account, in expanding the denominator of II, there is the same rejection of the terms that involve the T dv square and higher powers of frv. But we may obtain an expression without expanding the denominator of П, and, by merely multiplying every term of and it makes very little difference in the result, whether we use this, or that form of the differential equation which arises on subtituting instead of II its former value (1. 9.). |