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Suppose the plane of x, y, to represent the plane of the ecliptic, and the longitudes of bodies to be measured from the




line T; then, drawing L perpendicular to the plane of the ecliptic, the angle & TI is the longitude of L, the angle LTI is its latitude, TL is the radius, and Tl, the projected radius, is the curtate distance, (see Astron. p. 273.).

Let Tp and the angle & TI=v, and let the tangent of latitude (LTI) be s, then

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and, accordingly, instead of determining the body's place by x, y, and z, we may determine it, by p, v, and s, or by r, v and s.

This change in the conditions of the body's place, will render a change necessary in the mode of expressing the forces. The forces X, Y, acting in the directions of x and y, must be re




solved into others acting in the direction of p, and in a direction perpendicular to p.

The three differential equations of p. 8, will also be affected by this change in the mode of determining the body's place. They will lose their similarity, or cease to be symmetrical: three other equations will arise, but not symmetrical equations.

Since we know the values of x, y, z, in terms of p, v, and s, the transformation of the differential equations of p. 8, into others, is a mere matter of calculation; and of no difficulty, since we are guided in it by this property, namely, that

X cos. v + Y sin. v = force in the direction of T1 = P, and Y cos. v-X sin. v = force in the perpend. direction = ±T.

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dv. p. sin. པ

dp.sin.v + dv.p.cos. v,

d2x=d2p. cos. v − 2dv.dp.sin.v-p.dv2. cos. v-p. dv. sin. v, dy=d2p.sin. v+2dv.dp.cos.v-p. dv2. sin. vtp .d2 v. cos. v;

.. d'x. cos. v + ď2y. sin. v = dp — p.dv3,

d2y. cos.v d2 x. sin. v = 2 dv.dp+p.d2 v.

Hence, writing S instead of Z, and ps instead of z in the last equation, we have these three new equations,

d2 ppd v2 + P. dt = 0............[4]*, 2dvdp + pd2 v±T.dť2 = 0............ [5], d(ps) + S. dt = 0............ [6],

in which P, T, and S, represent the results of any number of

* These equations are the same as those which Dalembert has inserted in the 6th Volume of his Opuscules.

forces that act upon the body L; the first in directions parallel to p, the second in directions perpendicular to the former, and the third in directions perpendicular to the plane of the two first forces.

These three new equations are not like the equations [1], [2], [3], of p. 8, symmetrical, but, with regard to form, are totally unconnected, the one with the other; they possess, however, this advantage, that, when solved, they would immediately exhibit the values of p, v, and s, which are the quantities requisite, in Astronomical enquiries, to be known.

But even these last forms of the equations, although they possess advantages over the first, require some farther modification; and for the following reason. The first equation involves p, v, and t (P being some function of p and v): now a curve, in order that it may be traced out, or that its properties may be investigated, requires an equation expressing the relation either between its rectangular co-ordinates x and y, or between its radius vector, such as p, and ́an angle, such as v, contained between p and some line given in position. The first equation then, if integrated and solved, would not define the nature of the curve, since t, the time, would be involved in it: t, therefore, must be eliminated; and its elimination will be the object of a succeeding transformation.

But, as we shall hereafter see, the process will not terminate with that transformation. There will remain to be made another step; a very short one, indeed, consisting merely in substituting, instead of p and its functions,


and its corresponding functions.

This substitution, beyond what could be presumed from any antecedent reasons, is eminently useful in abridging the process of calculation: and was, probably, rather happily hit on, than found by any scientific clue *.

* In tracing the connection of the successive transformations, it will be thought, perhaps, that we have rather made a way than found one. It is, indeed, almost necessary, and certainly it is very commodious, to establish, between methods so difficult as those of Physical Astronomy,


If we were immediately to press forward to those most commodious and perfect forms, which the ingenuity and labour of Mathematicians have given to the Differential Equations of Motion, we should conduct the Student, in the out-set of his career, over too extended a field of apparently barren speculation. It is better to stop for a while and endeavour to collect some useful results. And this we shall be enabled to do by investing the preceding equations with the conditions that obtain in nature, that is, by substituting, instead of the general symbols P, T, and S, the expressions of those forces by which bodies in the Planetary system are mutually drawn towards each other.

The deduction of these results will be the object of the succeeding Chapters; but, in the one that immediately follows, the result obtained is altogether independent of the Law of Centripetal Forces.

an artificial connection, when no natural one exists. And scarcely any natural one exists. The approved methods of science are very different, in their form, from those which their inventors first exhibited, and still more different, probably, from those which were first investigated. Their present compactness and neatness is the fruit of numerous trials and experiments, of which the traces are not preserved.


Consequences that follow from the Differential Equations of Motion when the Forces acting on a Body in motion are Centripetal, or are directed to one Point only: Kepler's Law of the Equable Description of Areas demonstrated. Variation of the Velocity. The Equable Description of Areas necessarily disturbed, when the Body is acted on by Forces, some of which are not directed to the same Point or Centre.

IN the equations (4), (5), (6), of p. 10, the sole condition regulating the forces P, T, S is, that they should act on the body L in directions respectively parallel to T, perpendicular to LI, and parallel to LI. Let us now suppose the forces which act on L to act (previously to any resolution of them) solely in the direction of LT. In this case there can be no force to draw the body





out of the plane in which it once moves. It is not therefore essential (see p. 6,) to consider any other plane than that of the

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