We e must, for a time, disregard the peculiar nature of each element and find out or feign conditions of general resemblance. Now, this may be effected on the principles laid down in pp. 39, &c. The arbitrary quantities introduced by integration are to be considered as the elements of the orbit described, and, consequently, the differentials of such quantities will represent the variations of the elements. The differentials may, as in other cases, be found by the ordinary rules; but there is, in the subject we are now speaking of, some difficulty in reducing them to similar formulæ. Mere similarity, indeed, is in itself not necessarily an advantage: but the similar formulæ just alluded to will be found most convenient; both for the making of general inferences and for illustration by examples. This is the mere suggestion of the plan which will be more fully developed in the ensuing Chapter. CHAP. XXI. ON THE VARIATIONS, PERIODICAL AND SECULAR, OF THE ELEMENTS OF THE ORBITS OF PLANETS. Principle of the Method for determining the Variations of the Element's of a Planet's Orbit. The Elements viewed as the Arbitrary Quantities introduced by the Integration of the Differential Equations of Motion, or as their Functions. Expressions for the Variations of the Mean Distance, the Eccentricity and the Longitude of the Perihelion: the Variation of the Eccentricity expressed by means of partial Differential Coefficients of the Quantity (R) dependent on the Disturbing Force: the same Form of Expression extended to the Variations of the other Elements. The Origin and the Authors of these Ex pressions. IF F we revert to pp. 39, &c. we shall perceive that, according to the analytical view there taken of the subject, the arbitrary quantities, introduced by the integration of the differential equations, are either the elements of a planet's orbit, or functions of those elements. They are the elements themselves, when the equations are expressed in terms of the projected radius vector, the longitude and the tangent of latitude: that is, when the three equations (abstracting the disturbing forces), are and functions of the elements, when, by means of the rectangular co-ordinates x, y, and z, the equations are thus symmetrically expressed, Both these sets of equations (for they may be mutually derived, the one from the other) belong to the elliptical system, consisting only of two bodies, and in which there is no disturbing force. As we have already seen (pp. 53, &c.) the elements of the elliptical system, the axis major, eccentricity, &c. are invariable. The arbitrary quantities, introduced by the integration of the differential equations, are constant arbitrary quantities; and, whilst the equations retain the preceding form, they are capable of an exact integration. But there is no case in the planetary theory to which the preceding equations exactly apply. The elliptical laws of form and revolution are never strictly observed: there is always, more or less, some disturbance: and the preceding equations, before they are applicable to the planetary theory, require the addition of certain small terms dependent on the disturbing force. For instance, the equation of 1. 1. requires the addition of dx d R and the two other equations require, respectively, the addition of are very small, and therefore the integration of the differential equations, from which these terms are rescinded, must be nearly the integration of the equations when they are complete. The constant arbitrary quantities also, determined for the former equations, although they cannot remain the same, or retain their property of invariability, yet they cannot be subject to any except small variations, since the two sets of equations differ only by very small terms such as &c. d R The mean distance, for instance, if it were possible to determine it by the integration of the complete equations, could only differ, by a very small quantity, from that value of the mean distance which results from the actual integration of the imperfect equations of p. 376. This small difference, whatever be its expression, between the two mean distances, must be dependent on the disturbing force; for, the two sets of equations differ only by those small terms which would be nothing were there no disturbing force. Hence, (and it was by reasoning nearly in this way that the method was arrived at) the expressions for the constant arbitrary quantities resulting from the integration of the elliptical equations may be assumed as the expressions for the variable arbitrary quantities, on the condition of determining the variations of the latter from the differences between the two sets of equations. For, neither do the arbitrary quantities vary, nor do the equations differ, except by reason of the disturbing force. Suppose, then, a to be an arbitrary quantity, and that, by the integration of the elliptical equations, or of an equation resulting from their combination, we obtain an equation of the first order, such as V = a. V involving, or being a function of, x, y, z, But if d V = 0. a be assumed an integral equation of an equation formed by combining the equations, then in dV=da, we cannot, as before, substitute d2r d 12' -y instead of, &c. since the values of d 12' instead of d2 x d2y, &c. dt' di2' are different, as it is plain from the equations, or, as we may at once infer from this consideration; namely, that, the forces in the two cases are different, and forces are expounded by the second differentials or fluxions of quantities (see Preface to Principles of Anal. Calc. pp. 5, 6.). d2 x ferent values in the two cases, and, if we suppose the symbol d to denote the effect of the disturbing force, the first equation may |