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proximating to the value of u depends on the minuteness of

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The object of this Chapter has been said to be the integration of the equation

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which cannot generally be accomplished; but we have arrived at this important result, namely, the practicability of integrating it, if II could be represented by a series of the sines or cosines of multiple arcs. By that route then * we have a chance of arriving at our object; we have got something of a clue, and our next steps, it should seem, ought to be directed to the conversion of II into a series of such sines or cosines.

* This, however, is not the sole route by which the integration of the equation is to be arrived at. It would be attained if II could be re presented by K+nu, and in some cases, it may nearly be represented by such a quantity. For instance, if II should equal L+Mu+Nu", then, the orbit being nearly circular, and, consequently, u nearly = = (a being the mean distance,) we should have

u =


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by neglecting the higher powers of



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u; and the approximate

value for u would be of a similar form. Hence, ПI would be of the form K+nu, and consequently the differential equation would be

d2 u
~ ~ + (1 + n) u + K = 0,

the integration of which by the note of p. 101. is

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1 + n

This method, however, is a partial one; that is, it obtains only in particular circumstances, and, besides, its results are included amongst those of the general one, and in which II is represented by a series of cosines.

It is necessary, however, to examine certain circumstances that are adjacent to this part of the main route of investigation, before we proceed along it. These circumstances are peculiarities of solution, which, in certain predicaments, attach themselves to that method of approximation which has been described in the present Chapter. They are (and under this point of view we shall first consider them) analytical. But, in the application of the Calculus to the subject of this Treatise, they produce certain incongruities which vitiate that explanation of the Planetary Theory, which is founded on the principles of Physical Astronomy. It is necessary, therefore, to get rid of them: to shew why they vitiate, and how, by a modification of the Calculus, they may be made not to vitiate that explanation. The student, however, who, at this point of the enquiry, shall feel no inclination to attend to these peculiarities, may, in his first perusal, pass over the succeeding Chapter.


On certain Ambiguities of Analytical Expression that occur in the Problem of the Three Bodies; their Source and Remedy. A new Form for the Integral value of u from which the Arcs of Circles are excluded. Consideration on the Alteration which certain small Quantities may receive from the Process of Integration. Comparison between the Analytical Formula, and the Results of the Geometrical Method. Observations on the Ninth Section of the Principia.

Ir has already appeared (see p. 99.), that if, in the expression for the disturbing force, a term such as A. cos. mv should enter, the methods of integration would introduce into the expression of the value of u, this term

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If we make m=1, then, both the numerator and denominator are = O, and the term becomes

A x 0

which is an useless result. This is said to be a fault of calculation (faute du Calcul); but, if we examine the matter, it will appear that the above indefinite expression arises entirely from an extension of a rule. Thus (see p. 100.), the result of the integration gives

*There are many similar instances to be found in Analytical Science. The integral of is a case in point. The rule for finding

integral of f'da is

its integration cannot extend to that case, because in the enumeration of cases where the rule holds and is good, that particular one cannot be comprehended.

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where m is supposed to represent any number. But the value m = 1 must be excluded, and precisely for this reason; that, if we suppose the expression for the disturbing force to contain a term such as A. cos. v, the corresponding terms introduced into the value of u, by the process of integration, do not assume that form which belongs to them when m is expounded by any number 2, 3, 4, &c. and which therefore is restricted in its generality by the exception of the case in which m = 1.

In order to find the result of the integration in this particular case of m=1, we must substitute cos. v instead of cos. mv (see p. 98.); in which case, the quantity to be integrated will be vfcos. v. sin. v.dv sin. vfcos. v. dv


(see Trig. ed. 2. pp. 26. 36.)


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Here then, although not by deduction from the general formula, we have the expression for u when II is expounded by A cos. v, and the Calculus, if it can be said to have been faulty, is completely amended. We have, however, now to consider, whether any incongruity will be attached to the above peculiar value of u in its application to the Lunar Theory, or to the problem of the three bodies.

Now, between the general value of u (see p. 100.) and the preceding peculiar one, there is this notable difference; that, in the latter, the arc v appears without the sign of the sine; the consequence of which is, that 4 sin. v will, on the whole, by the increase of v,

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continually increase; and the value of u will at the end of any period be different from that which it was at the beginning, and more different, the greater the period. Preceding values of the radius vector

( = 1)

therefore cannot recur, and the curve traced out by the extremity of the radius vector cannot be of an oval form or reentering *. But it is clear from observation that the orbits of the planets are oval: their radii vectores, therefore, cannot be generally expounded by an expression such as the inverse of

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and consequently the Calculus is here also in fault, or the Planetary phenomena are not explicable on the preceding premises.

If we examine the process of p. 99. it will immediately appear that, should such a term as A cos. v enter into the composition of II, the process of integration must introduce the




v. sin. v. This point being certain and determined,

we are naturally led to enquire whether in expounding the disturbing force we must of necessity use such a term as A cos. v, or whether such a term finds its way into the expression for II by that peculiar method of approximate integration, to which, from want of ampler resources, the state of analytical science obliges us to resort.


'On voit par la que lorsque , renfermera des termes de cette espece, l'equation de l'orbite contiendra des angles v; et quelques petits que soient les termes ou ils entrent, ils peuvent donner les plus fortes corrections a la valeur de r, lorsqu'on supposera l'angle v d'un grand nombre de revolutions. Ainsi si l'on n'a rien negligé en determinant, on sera sûr l'orbite s'ecartera a la fin fort considerablement d'une ellipse et changera entierement de forme'. Clairaut, Theorie de la Lune, ed. 2. p. 11. See also Dalembert. Theorie de la Lune, pp. 30, 34, &c.

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