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elliptical value, (or rather its value in the undisturbed system) can only be known by the integration of the third equation,

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If therefore our object were scrupulously to compute the value of u, it would be necessary, after the approximations already pointed out to be made, to obtain, by the approximate integration of the third equation*, the value of s and to substitute it in the first equation.

In order, therefore, steadily to pursue the obvious method of successive corrections, it is necessary to deduce ds from the third equation, to substitute its value in the first, and then to deduce the value of u. But we shall be content, at present, with having pointed out the source of this new correction, of which however the detail and application, since it is small in degree, would not be very tedious. The design and scope of this Treatise call our attention to other points.

Of these the chief and most prominent is, the Progresssion of the Lunar Apogee; partly from its intrinsic importance in furnishing to Newton's system one of the best and most satisfactory tests of its truth: and partly from its historical importance; for, an error committed in the first computations of its quantity made those who had adopted Newton's system to waver in their belief of its truth, and revived, for the same reason, the spirits of the drooping Cartesians.

This subject of the Progression of the Lunar Apogee has been already, in several places (see pp. 146. 157. 181.) adverted to; and, in fact, the substance of the source of the error and of the means of correcting the error, are already in the possession of the Student.

* The integration of this equation, similar to that of the first would assign to ds an expression of this kind.

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It is merely for his convenience, and for the purpose of a complete elucidation, that we collect its several parcels and arrange them in order.

A second subject of enquiry, connected with the preceding, but, like it, digressive, relates to the determination of the Progression of the Lunar Apogee from the consideration of one force alone acting in the direction of the radius. This, if the progression be rightly determined on the condition of two forces, one in the direction of the radius, the other tangential, may be thought a futile enquiry; and, indeed, it deserves to be considered solely by reason of a sort of historical importance attached to it. Since some mathematicians, fancying themselves treading on the very footsteps of Newton, have sought for the quantity of the progression solely on the principles of the ninth Section.

These enquiries, if the main drift of the Treatise were merely the determination of the place of the disturbed planet, are not essential. And as, under any point of view, they partake somewhat of the nature of digressions, the Student will have the power of disregarding them as such, by passing over the next Chapter, which may be considered as separately assigned to them.


The Method of determining the Progression of the Apsides in the simplest Case of the Problem of the Three Bodies. Clairaut's Analogous Method for determining the Progression of the Lunar Apogee. His first Erroneous Result. Its Cause, and the Means of correcting it. Quantity of the Progression computed from the Condition of a Sole Disturbing Force acting in the Direction of the Radius Vector. Remarkable Result obtained by the first Integration of the Differential Equation. Dalembert's Method of Indeterminate Coefficients, for finding the Value of the Inverse of the Radius Vector, adopted by Thomas Simpson and Laplace.

THE simple instance of p. 109, &c., and which indeed is that which Clairaut (Theorie de la Lune, ed. 2. pp. 13, &c.) uses, will serve to illustrate that author's method of determining the Progression.

The general equation (see p. 109.) for determining u, in the Problem of the Three Bodies, is

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if we make μ = 1, and suppose the disturbing force to act solely in the direction of the radius vector and to be proportional to the inverse of its cube, we shall have (see p. 109.)

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This equation, if we make m' = 0 (which in fact is to suppose that there is no disturbing force), becomes the equation belonging to the elliptical system, and its integral determining u is of this form

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Clairaut, (see Theorie de la Lune, p. 13.) in order to verify the supposition, substitutes the assumed value of u in the differential equation; then, after the method described in pp. 99, &c. he integrates that equation, and compares the resulting value of with the assumed; the former is

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which, compared with the latter, will give rise to three equations for determining the three arbitrary quantities p, e and c: these equations are

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= 1.

From these three equations will result three values for p, e and C, such as must make the assumed and deduced values of u perfectly to coincide.

The third is the important equation: from that we derive

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which is not the approximate but the exact equation for the radius vector of a body, acted on by a force compounded of two parts, one varying inversely as the square, the other inversely as the cube of the distance, and both, strictly speaking, centripetal, and not perturbative of the equal description of areas; although the latter, from analogy of the language used on these occasions, may be termed a disturbing force *.

The preceding equation (determining the value of r) although similar to, is not, in fact, the equation to an ellipse. But, after certain conventions, such as have been explained in pp. 119, &c. it will serve to represent the radius of a moveable ellipse; moveable in such a manner, that its axis-major revolves round the focus, as round a fixed point, with an angular velocity which is to that of the body revolving in the ellipse, as

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This angular motion of the axis is, in other words, the progression of the apsides, which are its extremities; or, in the case of

By the discoveries of Kepler the orbits of the planets appeared to be elliptical, and when afterwards they were found not to be strictly so, mathematicians were still inclined to view the ellipse as the natural curve, and consequently would term the peculiar law of force producing it, the natural law of force: other forces therefore which disturbed the elliptical form would be termed disturbing forces. Chaque planete decriroit naturellement une ellipse si elle n'etoit attireé que par le corps autour du quel elle tourne,' says Lalande, (Astron. tom. III. p. 596.) But, it is easy to see, these are merely the denominations of a conventional language.

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