CHAP. XI.

CHANGE OF INCLINATIONS.

323

and is therefore calculable by spherical trigonometry, when the angle R Sr or the momentary recess of the node is known, and also the inclination of the planes of the orbits to each other. We perceive, then, that between the momentary change of inclination, and the momentary recess of the node there exists an intimate relation, and that the research of the one is in fact bound up in that of the other. This may be, perhaps, made clearer, by considering the orbit of M to be not merely an imaginary line, but an actual circular or elliptic hoop of some rigid material, without inertia, on which, as on a wire, the body P may slide as a bead. It is evident that the position of this hoop will be determined at any instant, by its inclination to the ground plane to which it is referred, and by the place of its intersection therewith, or node. It will also be determined by the momentary direction of P's motion, which (having no inertia) it must obey; and any change by which P should, in the next instant, alter its orbit, would be equivalent to a shifting, bodily, of the whole hoop, changing at once its inclination and nodes.

(508.) One immediate conclusion from what has been pointed out above, is that where the orbits, as in the case of the planetary system and the moon, are slightly inclined to one another, the momentary variations of the inclination are of an order much inferior in magnitude to those in the place of the node. This is evident on a mere inspection of our figure, the angle R Pr being, by reason of the small inclination of the planes SPR and R Sr, necessarily much smaller than the angle R Sr. In proportion as the planes of the orbits are brought to coincidence, a very trifling angular movement of Pp about PS as an axis will make a great variation in the situation of the point r, where its prolongation intersects the ground plane.

(509.) To pass from the momentary changes which take place in the relations of nature to the accumulated effects produced in considerable lapses of time by the continued action of the same causes, under circumstances

varied by these very effects, is the business of the integral calculus. Without going into any calculations, however, it will be easy for us to trace, by a few cases, the varying influence of differences of position of the disturbing and disturbed body with respect to each other and to the node, and from these to demonstrate the two leading features in this theory the periodic nature of the change and re-establishment of the original inclinations, and the small limits within which these changes are confined.

(510.) Case 1..

When the disturbing body M is situated in a direction perpendicular to the line of nodes, or the nodes are in quadrature with it: M being the disturbing body, and S N the line of nodes, the disturbing force will act at P, in the direction P K ; being a pulling force when P is in any part of the semicircle H A N, and a pushing force in the whole of the opposite semicircle. And it is easily seen that this force is greatest at A and B, and evanescent at H and N. Hence, in the whole semicircle H A, Pq will lie below Pp, and being produced backwards in the quadrant H A, and forwards in A N, will meet the circle Sb Na in the plane of M's orbit, in points behind the nodes SN, the nodes being retrograde in both cases. But the new inclination of the disturbed orbit is, in the former case, P x A, which is less than PH a; and in the latter, Pya, which is greater than PN a. In the other semicircle the direction of the disturbing force is changed; but that of the motion, with respect to the plane of M's orbit, being also in each quadrant reversed, the

CHAP. XI.

MOTION OF THE NODES.

325 same variations of node and inclination will be caused. In this situation of M, then, the nodes recede during every part of the revolution of P, but the inclination diminishes throughout the quadrant S A, increases again by the same identical degrees in the quadrant A N, decreases throughout the quadrant Nb, and is finally restored to its pristine value at S. On the average of a revolution of P, supposing M unmoved, the nodes will have retrograded with their utmost speed, but the inclination will remain unaltered.

(511.) Case 2.- Suppose the disturbing body now to be fixed in the line of nodes, or the nodes to be in syzygy, as in the annexed figure. In this situation the direction of the disturbing force, which is always parallel to S M, lies constantly in the plane of P's orbit, and therefore produces neither variation of inclination nor motion of nodes.

(512.) Case 3. Let us take now an intermediate situation of M, and indicating by the arrows the directions of the disturbing forces (which are pulling ones throughout all the semi-orbit which lies towards M, and pushing in the opposite,) it will readily appear that the reasoning of art. 510. will hold good in all that part of the orbit which lies between T and N, and between V and H, but that the effect will be reversed by the reversal of the direction of the motion with respect to the plane of M's orbit, in the intervals HT and N V. In these portions, however, the disturbing force is

feebler than in the others, being evanescent in the line of quadratures T V, and increasing to its maximum

in the syzygies a b. The nodes then will recede rapidly in the former intervals, and advance feebly in the latter; but since, as H approaches to a, the disturbing force, by acting obliquely to the plane of P's orbit, is again diminished in efficacy, still, on the average of a whole revolution, the nodes recede. On the other hand, the inclination will now diminish during the motion of P from T to c, a point 90° distant from the node, while it increases not only during its whole motion over the quadrant c N, but also in the rest of its half revolution N V, and so for the other half. There will, therefore, be an uncompensated increase of inclination in this position of M, on the average of a whole revolution.

(513.) But this increase is converted into diminution when the line of nodes stands on the other side of S M, or in the quadrants V b, T a; and still regarding M as fixed, and supposing that the change of circumstances arises not from the motion of M but from that of the node, it is evident that so soon as the line of nodes in its retrograde motion has got past a, the circumstances will be all exactly reversed, and the inclination will again be augmented in each revolution by the very same steps taken in reverse order by which it before diminished. On the average, therefore, of a WHOLE REVOLUTION OF THE NODE, the inclination will be restored to its original state. In fact, so far as the mean or average effect on the inclination is concerned,

CHAP. XI. STABILITY OF THE INCLINATIONS.

327

instead of supposing M fixed in one position, we might conceive it at every instant divided into four equal parts, and placed at equal angles on either side of the line of nodes, in which case it is evident that the effect of two of the parts would be to precisely annihilate that of the others in each revolution of P.

(514.) In what is said, we have supposed M at rest; but the same conclusion, as to the mean and final results, holds good if it be supposed in motion: for in the course of a revolution of the nodes, which, owing to the extreme smallness of their motion, in the case of the planets, is of immense length, amounting, in most cases, to several hundred centuries, and in that of the moon is not less than 237 lunations, the disturbing body M is presented by its own motion, over and over again, in every variety of situation to the line of nodes. Before the node can have materially changed its position, M has performed a complete revolution, and is restored to its place; so that, in fact (that small difference excepted which arises from the recess of the node in one synodical revolution of M), we may regard it as occupying at every instant every point of its orbit, or rather as having its mass distributed uniformly like a solid ring over its whole circumference. Thus the compensation which we have shown would take place in a whole revolution of the node, does, in fact, take place in every synodic period of M, that minute difference only excepted which is due to the cause just mentioned. This difference, then, and not the whole disturbing effect of M, is what produces the effective variation of the inclinations, whether of the lunar or planetary orbits; and this difference, which remains uncompensated by the motion of M, is in its turn compensated by the motion of the node during its whole revolution.

(515.) It is clear, therefore, that the total variation of the planetary inclinations must be comprised within very narrow limits indeed. Geometers have accordingly demonstrated, by an accurate analysis of all the circumstances, and an exact estimation of the acting forces,