The inclination, of nodes passes

in consequence of the regression of the node), and in the other half revolution, the same changes in every respect take place in the same order. therefore, is greatest when the line through the sun, or coincides with the line of syzygy; and is least when the line of nodes coincides with the line of quadratures; since it is constantly diminishing while we are going from the former state to the latter, and constantly increasing while we are going from the latter state to the former. This is the principal irregularity in the inclination of the moon's orbit; all the others are very small.

(204.) The line of nodes is constantly regressing at every revolution of the moon, except when the line of nodes passes through the sun. The annual motion which we might at first expect it to have, is somewhat diminished by the circumstance, that the rapid regression of the line of nodes, when in the position in which the greatest effect is produced, carries it from the line of quadratures more swiftly than the sun's progressive motion only, by making the line of quadratures to progress, would separate them. But as the line of nodes never progresses, the diminution of the motion of the line of nodes occasioned thus, is very much less than the increase of the motion of the line of apses (107.). Also, as the force acting on opposite points of the orbit, tends to produce effects of the same kind, there is no irregularity similar to that explained in (106.). Hence the actual regression of the line of nodes, though a little less than might at first be expected, differs from that regres

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sion by a much smaller quantity than that, by which the actual motion of the line of apses differs from the motion which at first we might expect it to have. The line of nodes revolves completely round in something more than nineteen years.

(205.) The effect of the irregularity in the regression of the nodes, and the effect of the alternate increase and diminution of the inclination, are blended into one inequality of latitude, which depends on the sun's longitude, the longitude of the moon's node, and the moon's longitude. This inequality was discovered (from observation) by Tycho Brahe, about A.D. 1590. It may be considered to bear the same relation to the inclination which the evection bears to the excentricity; and, like the evection in longitude, it is the greatest of the inequalities in latitude. It is, however, much less than the evection; its greatest effect on the moon's latitude being about 8', by which the mean inclination is sometimes increased and sometimes diminished.

(206.) There are other small inequalities in the moon's latitude, arising partly from the changes in the node and inclination, which take place several times in the course of each revolution (200.), &c.; partly from the excentricity of the orbits of the moon and the earth, partly from the distortion accompanying the variation, and partly from the variability of the inclination itself. We shall not, however, delay ourselves with the explanation of all these terms.

(207.) We shall now proceed with the disturbance of the planets in latitude.

In this inquiry it is always best to take the orbit of the disturbing planet for the plane of reference. Now let us first consider the case of Mercury or Venus disturbed by Jupiter. In this case, Jupiter revolving in a long time round the sun, which is the central body to Mercury or Venus, produces exactly the same effect as the sun revolving (or appearing to revolve) round the earth, which is the central body to the moon. The disturbing force of Jupiter, therefore, produces a regression of the nodes of the orbits of Mercury and Venus on Jupiter's orbit; and an irregularity in the motion of each node, and an alteration in the inclination, whose effects might be combined into one: and this is the only inequality in their latitude, produced by Jupiter, whose effects are sensible.

(208.) The other inequalities in latitude depending on the relative position of the planets, possess no particular interest; and a general notion of them may be formed from the remarks in the discussion of the motion of the moon's node. One case, however, may be easily understood. When an exterior planet is disturbed by the attraction of an interior planet, whose distance from the sun is less than half the distance of the exterior planet, and whose periodic time is much shorter, then the exterior planet is always farther from the interior planet than the sun is, and therefore, by (195.), there is a disturbing force urging it from the plane of reference when the planets are in conjunction, and to it when they are in opposition; and thus the exterior planet is pushed up and down for every con

junction of the two planets. The disturbance, however, is nothing when the exterior planet is at the line of its nodes (195.).

(209.) The near commensurability of periodic times, which so strikingly affects the major axis, the excentricity, and the place of perihelion, produces also considerable effects on the node and the inclination. The reasoning of (175.) and (176.) will in every respect apply to this case: the greatest effect is produced, both on the motion of the node and on the change of inclination, when the planets are in conjunction: the gradual alteration of the point of conjunction produces a gradual alteration of these effects, which, however (in such a case as that of Jupiter and Saturn), is partially counteracted by the gradual change of the other points of conjunction: the uncompensated part, however, may, in many years, produce a very sensible irregularity in the elements. If we put the words line of nodes for line of apses, and inclination for excentricity, the whole of the reasoning in (175.), &c., will apply almost without alteration.

(210.) For the secular variation of the position of the orbit, the following considerations seem sufficient. In the long run, the disturbed planet has been at every one point of its orbit a great number of times, while the disturbing planet has been at almost every part of its orbit. The disturbing force is always the difference of the forces which act on the sun and on the disturbed planet. As the disturbing planet, in these various positions, acts upon the sun in all directions in

the plane of its orbit, its effect on the sun may be wholly neglected; and then it is easy to see that, whether the disturbing planet be exterior or interior to the other, the combined effect of the forces in all these points on the disturbed planet at one point, is to pull it from its orbit towards the plane of the disturbing planet's orbit. (This depends upon the circumstance that the force is greatest when the disturbing planet is nearest.) Consequently, by (192.), the line of nodes of the disturbed planet's orbit on the disturbing planet's orbit, in the long run, always regresses. If the orbits are circular, there is no alteration of the inclination, since, at points equally distant from the highest point, there is the same force on the disturbed planet; and, therefore, by (192.), the inclination is increased at one time, and diminished as much at another. If the orbits are elliptic, one point may be found where the effect of the force on the inclination is greater than at any other, and the whole effect on the inclination will be similar to that.

(211.) In stating that the nodes always regress in the long run, the reader must be careful to restrict this expression to the sense of regressing on the orbit of the disturbing planet. It may happen that on another orbit they will appear to progress. Thus the nodes of Jupiter's orbit are made to regress on Saturn's orbit by Saturn's disturbing force. The nodes of these orbits on the earth's orbit are not very widely separated : but the inclination of Saturn's orbit is greater than that of Jupiter's. If we trace these on a celestial globe, we