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lities: the elliptic inequality being reckoned of the first; of a superior order it certainly is: for, its coefficient is 6° 18', whereas the coefficients of the Variation, Evection, and Annual Equation, are 35′ 46′′, 1° 20′ 29′′, and 11′ 11′′ respectively. But these inequalities do not exclusively occupy the second order: for, of the same order, (always determining the order by the magnitude of the coefficient) is the Second Elliptic Inequality, as it may be called, of which, A being the Moon's Anomaly, 2 A is the argument, and the coefficient (the greatest value of the equation) 12′. There is also another inequality of the same order denominated the Reduction: which, like the two elliptic inequalities, is almost enentirely independent of the disturbing force. The two latter principally depend on, or are derived from, the elliptic form of the orbit; whilst the reduction is owing to the inclination of the plane of the orbit to that of the ecliptic.

There are then five inequalities of the second order, and one of the first; and the true longitude expressed by the six terms that are their exponents, would be (reckoning the anomalies from perigee,)

v = nt + (6° 17′ 54′) sin. A,

Second Inequality...



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+ 1o. 20′. 29′′ sin. [2( D — O)— A],

Annual Equation......+ 11'. 11". sin. a,

7'.31". sin. 2 dist. D from N,

a representing the Sun's Anomaly.

These are the principal terms of the series expressing the value of v. If we continue the examination of the terms, we may select and arrange, into a third class or order, fifteen other terms expounding inequalities: of which the arguments would be

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After this third class we may carry the approximation still farther and form a fourth; and, as it has been observed (p. 234.)

there can be no end, in a merely mathematical view of the subject, of the terms composing the value of v; or, to use a different phraseology, an infinite number of equations result from theory to be applied as corrections to the mean longitude for the finding of the true. Of these it is sufficient to retain twenty-eight, the others being rejected; the sole rule and guide for rejection being their ascertained or computed minuteness *.

The coefficients of the terms (which are the greatest values of those terms) are constant. But the Moon, and Earth, the places of their apsides, and nodes, the inclination of the planes of their orbits; or, what technically is so called, the Configuration of the Earth, Moon and Sun, continually varying, the arguments which depend on such configuration, must continually vary: they must change from day to day. The Moon's place, therefore, if assigned on the principles of Physical Astronomy, would require every day the computation of nearly thirty terms (such as the terms of p. 229.). This would be very laborious. But in this, as in like cases, the labour, although it must always remain considerable, is lessened by the construction of Lunar Tables ↑.

Lunar Tables are not constructed solely by means of Physical Astronomy; nor, in fact, do they essentially require its aid. Three of the principal equations were determined, their coefficients as well as their arguments, long before Newton's discoveries. As they were determined so might the other equations, although the requisite labour of computation and observation would have been very great. Theory brings us sooner to the proposed end. It gives both the greatest values of the equations and their arguments in furnishing the latter it chiefly tends to improve the Lunar Tables; for the coefficients are most accurately determined by observation.

* We cannot have a surer guide than the computed exactness: but it will be oftentimes easy to see that terms, when they are multiplied by certain powers and products of the eccentricities and inclinations, must become too minute for any practical purposes of exactness. In such cases a formal computation, (oftentimes a troublesome one) may be dispensed with.

† See Mason's Lunar Tables: the Tables in the third Volume of Vince's Astronomy: and Tables de la Lune, par Burg.

The arguments of equations that have, like the Variation and Evection, short periods, may be determined by the scientific examination of observations alone. There is no great difficulty in this: the difficulty is to detect, without the aid of theory, inequalities of long periods. Take, for instance, that equation which was discovered by Laplace, and by which, within these few years, the Lunar Tables have been improved. It is at least problematical whether, by mere observation alone, this equation, whose period is one hundred and eighty-five years, would ever have been detected. The same may be said concerning the Secular Equation (see p. 182.) (in fact, a periodical equation of an extremely long period) discovered by the same Author. Previously to their being discovered, Astronomers were much embarrassed with certain Anomalies in the mean motion: for the secular inequality, and any inequality that slowly passes, by minute degrees, from its first increase or decrease, to its state of maximum or minimum, must necessarily blend itself with the mean motion, and perplex its determination.

The terms that represent the value of u (see pp. 180, &c.) expound the equations by which it is necessary to correct the Moon's mean parallax in order to obtain her true; and, in the present Chapter, we have given the mathematical explanation of the equations that serve to correct the Moon's mean longitude. It remains to find the Moon's latitude, and, since the latitude as well as the radius vector and longitude is affected by the disturbing force, to find the terms that expound the inequalities in latitude that is to be done by solving the differential equation [c] of p. 95. Now, setting aside the labour of computation, this is a matter of little difficulty, since the equation is similar to the equation which has been already integrated.

In the present Chapter we have spoken of the terms, that form the expression for the longitude, and of the corresponding equations, which they expound, as belonging to the Lunar Theory. That is, indeed, in Physical Astronomy, the theory of the greatest importance. But, it is plain, that inequalities and their corresponding equations, similar to the Lunar, will exist for every case of planetary disturbance for Venus disturbed by the action of the Earth, and


one of Jupiter's satellites disturbed by the action of another; and, with still stricter analogy, for any of the Satellites of Jupiter and Saturn disturbed by the Sun. There will be found to belong to these cases, equations the same, in the form of their arguments, as the Variation, Evection, and Annual Equation: not, indeed, so denominated, since the above terms have been appropriated, chiefly for historical reasons, to the Lunar Theory; nor always deserving to be distinguished on account of their magnitude: since the equations corresponding to the largest in the Lunar Theory are not necessarily the largest in an instance of the Planetary Theory: the magnitudes of the terms expounding equations depend, often as we have seen (seen pp. 236, &c.) on the proportion between the mean motions of the disturbed and disturbing body; which vary with the instance.




By this term is meant the equations that have similar arguments; 2.(?), for instance, in the case of Venus disturbed by Jupiter, is the argument corresponding to that of the Lunar Variation.


On the Integration of the Equation on which the Moon's Latitude depends. Formation of Equations correcting the Latitude. Regression of the Nodes. Secular Equation of the Regression.

Irs be the tangent of latitude,

y the tangent of the inclination of the plane of the orbit,

the longitude of the node,

g-1 the regression of the node *,

then, when no disturbing force acts, the finite equation

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When a disturbing force acts, of which the resolved parts are, P, S and T, the quantity s must be determined by the integration of the equation [c] of p. 95. which equation, reduced as the equation [b] was in page 138., is

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This equation is to be integrated exactly as the equation [b] has been in the preceding pages. The several parts, such as

To suppose, in this place, a regression, and to substitute an arbitrary quantity to represent it, is to anticipate a result; but the violation of the order of legitimate deduction is very slight, since it can be very easily shewn, (for what can be more simple than the reasonings of the tenth and eleventh Corollaries of the eleventh Section of the Principia), that the nodes cannot remain at rest: whether the motion be progressive or regressive does not affect the assumption of the term.

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