CHAP. XI. On the Corrections due to the Eccentricity of the Solar Orbit, and to the Inclination of the Plane of the Moon's Orbit. Method of deriving Corrections. Their Formula exhibited in a Table. The Error in the determination of the Lunar Apogee not removed by these Corrections. The deduction of Terms on which the Secular Equations of the Moon's Mean Longitude and of the Progression of the Apogee depend. THE immediate object of the introduction of the conditions of the inclination and the eccentricity of the Solar orbit, is, to obtain a more correct value of II in the differential equation and thence, by integration, a more exact value of u. It will easily be perceived, that the suppression of the abovementioned conditions must render the expression for II inaccurate. First, instead of being = a, (1 − e2) is = a (1 − e2 — y2). 1 1 Secondly, in u3, u' was made = whereas, if we suppose the Solar orbit to be like the Lunar, nearly an ellipse with a progressive apogee (1 denoting the progression), we ought to assume for w' an equation similar to the one for u (see pp. 140. 150.), and to make whereas, if the orbit be inclined, and y be the tangent of its in 1/ (1 + e2 + e . cos. cv), Lastly, the value of cos. 2 was obtained (see pp. 139. 150.) by equating the two expressions for the time which are both inaccurate, since they were computed from after that defective values had been substituted for u and u: on both accounts then cos. 2 w must be an inexact value, or will require two corrections. Corrected Values of sin. 2 w, cos. 2 w (see pp. 139. 150.). cos. 2v, *In the note to p. 38. the last term of the value of u is which is right when there is no disturbing force: but as (see pp. 110, &c.) the first approximate value of u obtained from the differential equation involves not cos. v, but cos. cv, so the first approximate value of s to be obtained by the integration of an equation similar to the former, will involve not sin. v, but sin. gv: and as, in the assumption of a value of u, the object is to assume one as near to the true value as we can, so the one in the text is assumed instead of that of p. 38. which belongs to the elliptical and undisturbed system. p. 150.) there must result a similar equation for the time: accordingly, since the denominator may be made 1. By equating now the two expressions for the time, we shall have, instead of the former expression for v', (see p. 151.) this v=mv - 2 me sin. cv + 3 me2 4 sin. 2cv 2 é sin, cv, is partly expressed by a funcin order therefore to get rid of in which, however, the value of tion of ', namely, 2 e sin. c'v': this term, multiply both sides of the preceding equation by ', and then by processes similar to those used in pp. 151, &c. * find by approximation, sin. cv': its value will be sin. c'v' sin. c'mv - mec' sin. (cv + c mv) If we now restore this value to the right hand side of the preceding equation, and (since is nearly = 1) make med = me, we shall have v = mv - 2 me sin. cv + 3 m 4 e2 sin. 2c v + 2 e. sin. cm v - 2mee [sin. (cv + cm v) + sin. (cv c'mv)] In this expression, the terms, after those in the first line, are an addition or increment to the value of v' arising from é cos. c'mv the increment of u'; or, if we wish to employ (which it is convenient to do) the symbol (d) of variations, the incremental terms may be considered as variations (dv) of arising from the variation (d) of u'. *See Trigonometry, p. 103. But if v be said to have a variation arising from du, it must also have one arising from u; for, the introduction of the inclination of the planes will add some small incremental terms to u, some therefore (see p. 150.) to the value of t, and accordingly, after equating the two values of t, some to the value of '. Now the integral taken and the denominators 4g, 2g - c made what they are equal to, 4 and 1 respectively, there will result very nearly t, * In deducing the coefficient of cos. co the expansion is continued till e3 cos. cv is included, which (see Trig. ed. 2. p. 53.) =cos. cv + cos 3 c v. The coefficient of cos. cv = −2e (1+322 + 513), and this 3 divided by 1+ 2+ 3+2 gives 372 2 -2e te(1–2), which, since cy2 is extremely small, is nearly 2e. And the other expressions = - are, in like manner, rendered more simple. In like manner, if we were to increase the value of u, by any (Q being very small) be increased, very nearly, by If, with these additional terms, we now equate the two values of t (see p. 140.) there will result + my sin. 2gvmey sin. (2 g v — cv) +2e sin. cm v-2 me é [sin. (cv +cm v) + sin. (cv — c'm v)] The first line in the preceding value of ', is its value (see p. 151.) when the inclination and the eccentricity of the Solar Orbit were supposed nothing. The second line is the increment of ', when the condition of the inclination of the plane of the Lunar Orbit is restored, or when The third line consists of terms that are incremental to ', when de' is supposed to be the eccentricity of the Sun's elliptical orbit, and when The fourth line is the increase to the value of v', when we either supply a deficient term to the value of u, or increase that u, by the successive integrations of the differential equation, acquires new terms: these cannot be known till after integration: the deficient terms of u are those which, for the sake of simplicity, are omitted in the assumed value of u. |