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The preceding expressions depend, and mainly, on the partial differentials of the quantity R. In that respect, therefore, they are similar; they are also, the formula for R being supposed to be established, of easy application.

The formula for R, which is requisite for such application, will be similar to that which was given in p. 279: but it must be extended so as to comprehend the terms that will arise on introducing the condition of the plane's inclination.

But the formulæ of the variations, although they possess a sort of symbolical similarity and are convenient for arithmetical computation, are derived by no simple processes. The fault, if it may be so called, does not lie entirely with the processes. The objects of investigation are abstruse. There is no immediate nor short connexion between the principle and Law of Gravity, and that, for instance, its result, which is called the Progression of the Perihelion.

It is also, in part at least, to be ascribed to the nature of the

* It must be recollected (see pp. 400, 401.) that R in these two expressions is not a function of the same quantities.

research, that its processes are not obvious, but rather indirect. What is the obvious and direct process which the consideration of the progression of the perihelion or of the variation of the axismajor suggests? The obvious and direct processes belong to much more simple subjects of enquiry; to the regression of the nodes, for instance, which is indeed one of the variations of the elements, and, therefore, an exception to what was said at p. 404. 11. 19, 20. But let any one, versed in these studies, and taking for his model the 30th Proposition of the third Book, or any other of the Propositions of the Principia, attempt similar constructions for the variations of the position and of the magnitude of the axis-major, and he will soon experience the very unaccommodating nature of these latter subjects. They are much too stubborn or subtle to yield to the ordinary modes of attack.

The elements of a planet's orbit being considered as certain arbitrary quantities introduced by the integration of the differential equations, the research of the variations of the elements immediately becomes a merely mathematical enquiry. The object of research, so far from directing the research, is altogether lost sight of. We come upon it all at once when the investi- ' gation is finished. If this be an imperfection in science it does not belong to the infancy of science.

As science advances, its processes become more compendious, and have less affinity, if we may so express ourselves, with their subjects.

The variation of the axis-major, which, by direct and obvious methods, is most difficult of approach, is, by the preceding methods, most easily investigated. Analogy would lead us to a different conclusion.

Newton, as it has been observed, did not treat of the variations of all the elements, nor did those mathematicians who formed, what may be called, the first set of his successors, and who endeavoured to establish the system of gravitation more firmly than its great founder had time to do. Clairaut, Dalembert, and Thomas Simpson determined the progression of the Lunar Apogee ; the first by the method described in Chapters IX, XIII: the two

latter by the method of indeterminate coefficients (see pp. 209, &c.) In determining the regression of the nodes and the variation of the inclination of the plane, they adopted, in fact, Newton's method than which, as an independent and simple method nothing can be better. The first step, indeed, towards the variaation of the parameters (which is now the method) was made by a contemporary of the above-mentioned mathematicians.

Euler*, in his Theoria Luna, p. 8, resolved the third differential equation into two others; one expounding the regression of the node, and the other the connected change in the plane's inclination. The invention, however, of the general formulæ belongs to the mathematicians of the second set, of which the most distinguished are Lagrange and Laplace. The former of these mathematicians deduced, in the Berlin Memoirs for 1774, that remarkable formula (see pp. 327, 380, 385.) from which it follows that the mean distances of the planets are subject to no secular variation from disturbing forces. Laplace arrived at this latter result by a different process: and the subject continued to be cultivated till Lagrange was enabled to express the variations of the elements by the formulæ of p. 404. This he did in the Memoirs of the Institute for 1809, by processes strictly analytical, but very long, and on principles not naturally, we may say, suggested by the subject of enquiry. Laplace in the supplement to the 10th Volume of the Mec. Cel. by more simple means, converted the formulæ he had already invented, into Lagrange's; and, in a Memoir inserted in the 9th Volume of the Ecole Polytechnique, M. Poisson has, and by a peculiar method, arrived at the same formulæ.

In the next Chapter we will deduce certain simple results from the preceding formulæ and illustrate the formulæ by examples. For that purpose it will be requisite to extend (see p. 279.) the expression for R in order to adapt it to their application. That, therefore, will be the first operation.

Mais Euler est le premier qui ait cherché a les determiner par l'analyse, ses formules etant de peu d'usage par leur complication', &c. Lagrange, Mem. Inst. 1809. p. 364.

CHAP. XXII.

Deduction of the constant Parts of the Development of R. Expressions for the Secular Variations of the Elements. Variations of the Eccentricities of the Orbits of Jupiter and Saturn. Theorem for shewing that their Eccentricities can neither increase nor decrease beyond certain Limits. Diminution of the Eccentricity of the Earth's Orbit. It is the Cause of the Acceleration of the Moon's Mean Motion. Its Value computed from the disturbing Forces of the Planets. Thence, the Secular Equation of the Moon's Acceleration computed. Variation of the Longitude of the Perihelion: sometimes a Progression, at other times a Regression. The Progressions of the Perihelia of Jupiter and Saturn computed. Variations of Inclination and of Node. Theorem for shewing that the Inclinations of the Planes of Orbits oscillate about a mean Inclination. The Mean Motions of Nodes, with reference to the Ecliptic, sometimes Progressive, at other times Regressive: but, with reference to the Orbit of the disturbing Planet, always Regressive. The Moon's Nodes. The Quantity of their Regression computed. Variation of the Obliquity of the Ecliptic: Progression of the Equinoxes; both caused by the disturbing Forces of the Planets: their Quantities computed. The Length of the Tropical Year affected by them.

THE value of R in Chapter XVII. is an incomplete value, because, in deducing it, the inclination of the plane was neglected or supposed equal nothing. If we restore that neglected condition, we shall have

R =

m'. (xx+yy + z z′)

m'

√ [x' —x)2 + 'y' − y)2 + (z' — z )2 ] °

--

Let x, x', be supposed to be measured along the intersection of the two orbits, y, y', in the plane of the orbit of m, and the angular distances, or longitudes v, v', in the planes of the respective orbits of the bodies m and m'; and, let the inclination of those

planes be; then, for the purpose of converting R into a function of r, r', v, v', &c. we have

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xx+yy=rr cos. v. cos. v'+r r'. sin. v. sin. v. cos.

=rr' (cos. v. cos. ' + sin. v. sin. v')—r r'. sin. v sin. v′ ( 1 cos. ), (Trig. pp. 26, 36.)

= rr'. cos. (v' —v) — 2 r r'. sin. v . sin. v .

v.sin.2

Φ 2

Again, the square of the denominator of the second term in the value of R equals

or

x2 + y2 + x22 + y22 + z'2 — 2 (x x' + yy),

p2 + p2—2rr cos. (vv)+4rr. sin. v sin. v. sin.2
1.2 2.

Hence, if we develop the second term in the value of R; (but,

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by reason of the smallness of sin2. not beyond the second

term), we shall have

2

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+

2 m'r r' sin2. sin. e. sin. v

√[r'2-2rrcos. (v'—v) + r2]' (r'2-2 rr' cos. (v' − v)+r2) +

In which expression the first and third terms are those which are given and expanded in Chapter XVII. The terms arising from the developments of the second and fourth are those which, arising from the inclination of the plane, are necessary to complete the value of R.

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