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course of investigation; and we will augment still more its digressive nature by briefly commenting on the method, by which Thomas Simpson and Laplace have obtained the coefficients of the general equation,

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The method which these two mathematicians use may be characterised as that of indeterminate coefficients. It was first suggested by Dalembert (Theorie de la Lune, pp. 107, &c.) who, however, does not adopt it, but employs for his practical solution, one of approximation and integration similar to that which has been described (pp. 137, &c.). The method of indeterminate coefficients Dalembert recommends as a good one, care being taken previously to ascertain the form of the series to be determined *; by which he means that the multiple arcs, or arguments (such as 2 v−2 m v, 2 v − 2 m v-cv, &c.) according to the cosines of which the series is to be arranged, must be previously determined. Now this caution is observed both by Thomas Simpson and Laplace. The former in his Miscellaneous Tracts, p. 148. first approximately integrates the differential equation in order to discover + the arguments or arcs, the cosines of which would be involved in the terms of the series for the inverse of the radius vector (u), and then assumes a series for u, the terms of which are the products of the cosines of the deduced arcs and of certain arbitrary quantities, such as B, C, &c.

* Cette maniere d'appliquer la methode des indeterminés a la solution d'une probleme dont il s'agit, est sans comparaison la plus courte et la plus facile de toutes, puis qu'elle ne demande ni integration ni aucun addresse de calcul.' p. 107. Again, 'Cette methode exige quelques precautions, pour ainsi dire, preliminaires; sçavoir, de prouver que la forme qu'on suppose a l'equation est en effet la seule qu'elle doive avoir. Or, j'ai cru qu'il etoit plus court de chercher directement cette forme en integrant rigoureusement et absolutement l'equation proposée,' &c. p. 109, &c.

† 'But, since the former operation is made, more with a view to discover the form of the series, than to be regarded for its exactness, I shall have no further reference thereto, but proceed to determine the several quantities e, B, C, &c. de novo, by a method somewhat different from that used above.' P. 148.


Now, if we revert to pp. 184, &c. it will appear that the ne cessity of correcting the coefficients of the terms of the series arose from II having been deduced from the elliptical and imperfect value of u. The corrections successively arise on restoring to u its deficient terms: they will, therefore, be of necessity superseded if the component parts of II be, in the first instance, deduced not from the elliptical value of u, but from that series which, with regard to its form at least, rightly represents its value. What will require to be done more than was done in pp. 185, &c. is the determination of the assumed arbitrary or indeterminate coefficients and for this purpose there will be an equal number of equations.

The erroneous determination of c arose, as we have seen, from the component parts of ПI having been deduced from the imperfect and elliptical value of u. That error, therefore, must necessarily be avoided by this method of Simpson, which, in the first instance, is founded on what may be viewed as a complete representation of the value of u; c, therefore, is determined with as much exactness as the method of approximation (for after all we are still thrown back on such methods) will admit of. And this Simpson states to be one of the advantages of his method*.

Laplace in his Mecanique Celeste, (tom. III. pp. 191,&c.) although, in the main, he follows Dalembert's suggested method, yet follows it not so closely as Simpson has done. He first, on the assumption of the elliptical value of u, deduces the values of the coefficients of the terms of the differential equation, and expresses them by means of the quantities m, e, e', c, &c. Observing then the forms of those terms that would constitute the increment to the elliptical value of u arising from the disturbing force, Laplace assumes (du representing the above-mentioned increment),

* It not only determines the motion of the apogee in the same manner, but utterly excludes, at the same time, all terms of that dangerous species (if I may so express myself) that have hitherto embarrassed the greatest mathematicians, and that would, after a great number of revolutions, entirely change the figure of the orbit.' Simpson's Tracts, Preface.

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The next step in Laplace's process is to correct the value of II, previously obtained on the ground of the elliptical value of u, by supposing u to vary, and its variation (du) to have that form which has been just assigned to it.

The last operation of Laplace's is to substitute in the differential equation which resulted from the previous operations (the coefficients of the terms being compounded of m, e, é, c', &c. and of Q, Q", Q" &c.) for u, this value

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thence will result an identical equation such as

A + B cos. cv +C cos. (2 v—2mv) + De. cos. (2v-2mv — cv) + &c. in which, A, B, C, &c. will be (to use a general term) functions of e, m,c, and of Q, Q", Q", &c. and, for the determining of these latter quantities, (for they being known the variation of u arising from the disturbing force will be known) there will be these equations,

A = 0,

B = 0,

C = 0,


(see Laplace, Mec. Cel. Partie 2de. Liv. VII. pp. 215, &c.)

From this brief account, besides for the reasons stated in pp. 209, 210, &c. it will appear that no error, nor any semblance of

* The additional terms due to the disturbing force have the same form in the differential as in the integral equation that assigns the value For instance, if P cos. pz be a term in the former, then

of u.

P 1

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error, in the determination of the progression of the apogee, similar to that which occurred in Clairaut's first Essays, can take place in this method.

Although the method of indeterminate coefficients is a sure and excellent one, yet it has not been adopted in these pages. Instead of it, we have employed another less scientific, perhaps, but more simple and obvious, more in unison with preceding methods and better suited to the purpose and plan of the Treatise. Nor are these latter advantages counterbalanced by any incorrectness. For by means of the Table and formulæ (see p. 175.) the method is capable of receiving a series of successive corrections.

We will now resume the main course of investigation, and proceed to the solution of the second equation (see p. 95.); thence we shall have t in terms of v, and consequently, the mean anomaly in terms of the true; but the solution depends (see p. 95.) on the tangential force T and on u. The value of this quantity u, therefore, requires to be known previously to the determination of the time. It is not, therefore, without reason that the equation [b] of p. 95. claims precedence of consideration and the deduction of the value of u is, perhaps, of not less importance for collateral purposes than for the obvious and direct one of determining the parallax.


Expression for the Time: first, when the Body revolving in a Circular Orbit is disturbed by the Action of a very distant Body. The Mean Longitude expressed in Terms of the True: the True thence expressed in Terms of the Mean by the Reversion of Series. The Introduction of Inequalities in the Mean Motion by the Disturbing Force: the Elliptic Inequality, the Variation: the greatest Value of the latter in an Orbit nearly Circular. Expression for the Differential of the Time in an Elliptical Orbit, the Disturbing Body revolving also in an Orbit of the same kind. The Expression integrated, and the Mean Longitude expressed in Terms of the True. Expression in this Case, of the Coefficient or greatest Value of the Variation. The Secular Equation of the Mean Motion, explanatory of the Acceleration of that Motion. Digression concerning the Properties and Uses of the Formula of Reversion. By means of that Formula the True Longitude expressed in Terms of the Mean: the Terms expound Inequalities: the greatest denominated the Variation, the Erection, the Annual Equation, the Reduction: Causes of their Magnitude. Lunar Tables, in what manner, improved by Theory.

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which expresses the differential of the time in a disturbed orbit, is reduced, when T the tangential disturbing force = 0, to (see p. 96.)

dv dt = hu'

and this latter (see p. 14.) is the analytical expression of Kepler's Law of the Equable description of Areas.

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