velocities then are to each other in a ratio compounded of these two proportions, that is in the ratio of

1:1+3a-4a-2b-2c,

which is evidently that of a greater to a less quantity. It is obvious also, from the constitution of the second term of this ratio, that the normal force is far more influential in producing this result than the tangential.

(711.) In the foregoing reasoning the sun has been regarded as fixed. Let us now suppose it in motion (in a circular orbit), then it is evident that at equal angles of elongation (of P from M seen from S), equal disturbing forces, both tangential and normal, will act: only the syzygies and quadratures, as well as the neutral points of the normal force, instead of being points fixed in longitude on the orbit of the moon, will advance on that orbit with a uniform angular motion equal to the angular motion of the sun. The cuspidated curves a, d1b, e, and a2 d2b2e2, fig. art. 708., will, therefore, no longer be re-entering curves; but each will have its cusps screwed round as it were in the direction of the sun's motion, so as to increase the angles between them in the ratio of the synodical to the sidereal revolution of the moon (art. 418.). And if, in like manner, the motions in these two curves, thus separately described by H, be compounded, the resulting curve, though still (loosely speaking) a species of oval, will not return into itself, but will make successive spiroidal convolutions about S, its farthest and nearest points being in the same ratio more than 90° asunder. And to this movement that of the moon herself will conform, describing a species of elliptic spiroid, having its least distances always in the line of syzygies and its greatest in that of quadratures. It is evident also, that, owing to the longer continued action of both forces, i. e. owing to the greater arc over which their intensities increase and decrease by equal steps, the branches of each curve between the cusps will be longer, and the cusps themselves will be more remote from S, and in the same degree will the dimensions of the resulting oval be enlarged, and with them the amount of the inequality in the moon's motion which they represent.

(712.) In the above reasoning the sun's distance is supposed so great, that the disturbing forces in the semi-orbit nearer to it shall not sensibly differ from those in the more remote. The sun, however, is actually nearer to the moon in conjunction than in opposition by about one two-hundredth part of its whole distance, and this suffices to give rise to a very sensible inequality (called the parallactic inequality) in the lunar motions, amounting to about 2' in its effect on the moon's longitude, and having for its period one synodical revolution or one lunation. As this inequality, though subordinate in the case of the moon to the great inequality of the variation with which it stands in connexion, becomes a prominent feature in the system of inequalities corresponding to it in the planetary perturbations (by reason of the very great variations of their distances from conjunction to opposition), it will be necessary to indicate what modifications this consideration will introduce into the forms of our focus curves, and of their superposed oval. Recurring then to our figures in art. 706, 707., and supposing the moon to set out from E, and the upper focus, in each curve from e, it is evident that the intercuspidal arcs e a, a d, in the one, and ep, pal, ld, in the other, being described under the influence of more powerful forces, will be greater than the arcs db, be, and dm, mbn, ne corresponding in the other half revo

h

lution.

f

The two extremities of these curves then, the initial and terminal places of e in each, will not meet, and the same

conclusion will hold respecting those of the compound oval in which the focus really revolves, which will, therefore, be as in the annexed figure. Thus, at the end of a complete lunation, the focus will have shifted its place from e to ƒ in a line parallel to the line of quadratures. The next revolution, and the next, the same thing would happen. Meanwhile, however, the sun has advanced in its orbit, and the line of quadratures has changed its situation by an equal angular motion. In consequence, the next terminal situation (g) of the forces will not lie in the line ef prolonged, but in a line parallel to the new situation of the line of quadratures, and this process continuing, will evidently give rise to a movement of circulation of the point e, round a mean situation in an annual period; and this, it is evident, is equivalent to an annual circulation of the central point of the compound oval itself, in a small orbit about its mean position S. Thus we see that no permanent and indefinite increase of excentricity can arise from this cause; which would be the case, however, but for the annual motion of the sun.

(713.) Inequalities precisely similar in principle to the variation and parallactic inequality of the moon, though greatly modified by the different relations of the dimensions of the orbits, prevail in all cases where planet disturbs planet. To what extent this modification is carried will be evident, if we cast our eyes on the examples given in art. 612., where it will be seen that the disturbing force in conjunction often exceeds that in opposition in a very high ratio, (being in the case of Neptune disturbing Uranus more than ten times as great). The effect will be, that the orbit described by the center of the compound oval about S, will be much greater relatively to the dimensions of that oval itself, than in the case of the moon. Bearing in mind the nature and import of this modification, we may proceed to enquire, apart from it, into the number and distribution of the undulations in the contour of the oval itself arising from the alternations of direction plus and minus of the disturbing forces in the course of a synodic revolution. But first it should be mentioned that, in the case of an exterior disturbed by an interior planet,

the disturbing body's angular motion exceeds that of the disturbed. Hence P, though advancing in its orbit, recedes relatively to the line of syzygies, or, which comes to the same thing, the neutral points of either force overtake it in succession, and each, as it comes up to it, gives rise to a cusp in the corresponding focus curve. The angles between the successive cusps will therefore be to the angles between the corresponding neutral points for a fixed position of M, in the same constant ratio of the synodic to the sidereal period of P, which however is now a ratio of less inequality. These angles then will be contracted in amplitude, and, for the same reason as before, the excursions of the focus will be diminished, and the more so the shorter the synodic revolution.

(714.) Since the cusps of either curve recur, in successive synodic revolutions in the same order, and at the same angular distances from each other, and from the line of conjunction, the same will be true of all the corresponding points in the curve resulting from their superposition. In that curve, every cusp, of either constituent, will give rise to a convexity, and every intercuspidal arc to a relative concavity. It is evident then that the compound curve or true path of the focus so resulting, but for the cause above mentioned, would return into itself, whenever the periodic times of the disturbing and disturbed bodies are com mensurate, because in that case the synodic period will also be commensurate with either, and the arc of longitude

intercepted between the sidereal place of any one conjunction, and that next following it, will be an aliquot part of 360°.

H H

In all other cases it would be a non-reentering, more or less undulating and more or less regular, spiroid, according to the number of cusps in each of the constituent curves (that is to say, according to the number of neutral points or changes of direction from inwards to outwards, or from accelerating to retarding, and vice versa, of the normal and tangential forces,) in a complete synodic revolution, and their distribution over the circumference.

(715.) With regard to these changes, it is necessary to distinguish three cases, in which the perturbations of planet by planet are very distinct in character. 1st. When the disturbing planet is exterior. In this case there are four neutral points of either force. Those of the tangential force occur at the syzygies, and at the points of the disturbed orbit (which we shall call points of equidistance), equidistant from the sun and the disturbing planet (at which points, as we have shown (art. 614.), the total disturbing force is always directed inwards towards the sun). Those of the normal force occur at points intermediate between these last mentioned points, and the syzygies, which, if the disturbing planet be very distant, hold nearly the situation they do in the lunar theory, i. e. considerably nearer the quadratures than the syzygies. In proportion as the distance of the disturbing planet diminishes, two of these points, viz. those nearest the syzygy, approach to each other, and to the syzygy, and in the extreme case, when the dimensions of the orbits are equal, coincide with it.

(716.) The second case is that in which the disturbing planet is interior to the disturbed, but at a distance from the sun greater than half that of the latter. In this case there are four neutral points of the tangential force, and only two of the normal. Those of the tangential force occur at the syzygies, and at the points of equidistance. The force retards the disturbed body from conjunction to the first such points after conjunction, accelerates it thence to the opposition, thence again retards it to the next point of equidistance, and finally again accelerates it up to the conjunction. As the disturbing orbit contracts in dimension the points of equi