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cnt, 2cnt, 2nt-2nmt, 2nt-2nmtcnt, &c.; or, making A to represent the Moon's mean anomaly, the arguments will be

A, 2A, 2(D − ☺), 2(D − ©) ·

A, &c.

But, it is plain, if we look merely to the mathematical or symbolical exhibition of terms, their number is infinite. Except, then, several should be eminent above the rest for their magnitude, or, to state the matter more correctly, except after certain terms, all succeeding terms should be so minute as not to be worth considering, the true longitude could not be computed either directly from the preceding series, or by its aid.

The fact is that all terms, saving about thirty, may, from their minuteness, be rejected; and, if we examine the series we shall perceive the causes of the minuteness; since, past a certain term, the coefficients involve the cubes of the eccentricities and inclination and their products.

The terms then to be retained and to expound equations, are considerably the greatest of the series, and their magnitude, or, rather, that of their coefficients depends, if we regard the differential equation subsisting between dv and dt, on two causes; the magnitude of that modification of the disturbing force which produces the inequality; and, the duration of its agency. Thus, if

dv = ndt + &c. + P. cos. pnt × ndt;

then, dv will, in a given element of time (dt), be the greater, the greater P is; P being a function of the disturbing force and other quantities such as the eccentricity, &c. The difference between the true longitude and the mean will continue to increase whilst P cos. pnt remains of the same sign. The whole excess, therefore, of the true above the mean longitude, will depend (P being given) on the length of time that P. cos. pnt continues of the same sign; and, that must depend on p: the smaller p, the greater will be the period of cos p n t passing from a given positive value to an equal negative one: the larger p, the quicker will be the transition from the positive to the negative values of cos. pnt.

This is to view the subject on certain intelligible grounds of

cause and effect: but, by the mathematical process, we may arrive, and in a more summary way, at the same result: for if

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and v will be the greater, the greater P is and the smaller p.

The Variation then, which is one of the principal Lunar Equations, must derive its magnitude from that of the modification of the disturbing force producing it: since, 2 nt - 2 nmt being its argument, the divisor 2 - 2 m introduced by integration is greater than 1. Or, in the other mode of considering the matter, we may say that the intensity of the disturbing force must be considerable, since the whole period of its action does not exceed fifteen days *.

If we apply the mathematical mode of estimating the magnitudes of the terms to the expression for dt, we shall easily see what are the kind of terms that give rise to considerable equations. Now the expression is

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* The exact period is 14,765294, or half a synodic period. During this, the variation passes through all its degrees of magnitude, positive as well as negative: the time, therefore, during which the Variation either continually augments, or continually diminishes the longitude, can be only one-fourth of a synodic period. In order to deduce this result, we may observe that sin. z passes through all its positive and negative values, whilst z passes from 0 to a value =360°: now, 2nt-2mnt=0, when t0: and, in order to find that value of t which makes 2nt 2 mnt 360°, we have, n denoting the Moon's motion in a given time (a day for instance),

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Here the last term involves du, the variation of u arising from the disturbing force; and this variation is obtained by integrating the equation,

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and if P cos. pv (see p. 101.) should be a term in II, then P. cos. p would be the corresponding term introduced by intep2


gration into the value of u. If p then should be nearly equal 1, there would be introduced, on that account, into the value of du, and consequently into that of d t, a term with a large coefficient. Now, if we examine the expression for du, we shall find the third term somewhat under the above predicament: the argument of that term is

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the term therefore in du that involves the argument (2-2 m-c)v must have received by the integration a small quantity as its divisor; when the integral of d t is taken, the second integration will introduce the divisor 2-2 m-c, which will not, however, much affect the value of the coefficient.

The term we are now speaking of expounds, in the Lunar Theory, the equation which is called the Evection, and which, together with the Variation (see Astron. Chap. XXXIV.) was discovered long before the rise of Physical Astronomy.

The second integration, we have just seen, introduces, if the term in dt be A. cos. (2 v 2 m v cv), a divisor 2-2m -C. That quantity differing little from 1 does not much alter the re



sulting coefficient-2m- but, it is plain, if in P

a term of dt, p should be very small, that the coefficient of the

resulting term would be considerably increased by integration: now the last term A. cos. c'mv in the value of a du (see p. 222.) is nearly in this predicament; for, since

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consequently the term A. cos. c'mr, which is very small in the expression for dt, will become of some magnitude


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A® é sin. dmv) in the integrated expression.

ć m

The term that we have been just considering expounds, in the Lunar Theory, the Annual Equation, (see Astronomy, Chap. XXXIV.) discovered, as the Variation and Evection were, long before the rise of Physical Astronomy.

Thus, the simple consideration of the divisors introduced by integration has helped us to the mathematical explanation of the magnitudes of two of the principal Lunar Equations. The magnitude of the Variation is derived from that of the modification of the disturbing force producing it. Almost the, whole of the tangential disturbing force is so expended. (see p.218.)

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The cause however of the magnitude of the annual equation is easily to be discerned without the aid of the mathematical explanation. It is owing to the duration of that modification of the disturbing force which produces it: which duration is half the period of a Solar Anomalistic Year *. The magnitude of the Evection arises not from any acceleration in the direction of the

* For reasons already stated in the note of p. 235. the whole period of the annual equation is to be determined from this equation, 360° = n. 's period;


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the annual equation, therefore, increases continually, or diminishes continually, during only half an Anomalistic year, the longitude.

tangent, but from alterations produced in the ellipse by that portion of the disturbing force which acts in the direction of the radius. There is an inequality due solely to the eccentricity of the orbit. The equation of the centre becomes greater by the ellipse becoming more eccentric. If therefore the disturbing force introduces any periodical change in the eccentricity, there will be a corresponding change in the equation of the centre; something superadded to, or taken away from that equation, which is independent of all disturbing force, and which is solely due to the original ellipse: and this periodical augmentation and diminution is a new Equation: it is, in the case we are considering, the Evection, arising from the disturbing force in the direction of the radius altering, what some mathematicians have been pleased to call, the Natural Centripetal Law. We see by this, why we must look for the magnitude of the Evection in the integration of the equation [b] of p. 95. which determines the radius vector. The merely mathematical cause of the magnitude of the term expounding the Evection consists, as we have seen in p. 236., in its receiving a small divisor (2 - 2 m — c)2 — 1.

The principle that has enabled us to assign the cause of the preceding Lunar, equations will serve to guide us in our search of others. The Annual Equation, as we have seen, although originating from a small modification of the disturbing force, yet becomes considerable by the accumulation of its effects. A slight modification of the disturbing force corresponds, in the differential equation, to a small coefficient: and the accumulation of effects depends on the period of the inequality. We must, therefore, in reducing the differential equation by the rejection of small terms, be careful to examine those terms that expound inequalities of long periods. The lengths of the periods may always be determined on the principles laid down in pp. 235. 237.

The three equations, the Variation, the Evection and the Annual Equation, have claims to particular consideration from their historical celebrity. They have received, unlike the other inequalities that depend on the disturbing force, a technical denomination. If they be viewed solely with regard to their magnitude, they will be found to belong to the second order of inequa

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