the arc a d be farther from perigee than b c, and therefore the force in it is greater, yet the predominance of effect here will not be very marked, and will moreover be partially neutralized by the small predominance of an opposite character in Y d over Z b. On the other hand, the arcs a Z, c Y are both larger in extent than either of the others, and the seats of action of forces doubly powerful. Their influence, therefore, will be of most importance, and their preponderance one over the other, (being opposite in their tendencies,) will decide the question whether on an average of the revolution, the excentricity shall increase or diminish. It is clear that the decision must be in favour of c Y, the apogeal arc, and, since in this the force is outwards and the moon receding from the earth, an increase of the excentricity will arise from its influence. A similar reasoning will, evidently, lead to the same conclusion were the apogee and perigee to change places, for the directions of P's motion as to approach and recess to S will be indeed reversed, but at the same time the dominant forces will have changed sides, and the arc a AZ will now give the character to the result. But when Z lies between A and E, as the reader may easily satisfy himself, the case will be altogether different, and the reverse conclusion will obtain. Hence the changes of excentricity emergent on the average of single revolutions from the action of the normal force will be as represented by the signs and — in the figure above annexed.

(690.) Let us next consider the effect of the tangential force. This retards P in the quadrants AD, BE, and

in the middle of cach quadrant, the tangential force is at its

maximum. Now, in the other quadrants, E A and D B, the change from perigeal to apogeal vicinity takes place, and the tangential force, however powerful, has its effect annulled by situation (art. 670.), and this happens more or less nearly about the points where the force is a maximum. These quadrants, then, are far less influential on the total result, so that the character of that result will be decided by the predominance of one or other of the former quadrants, and will lean to that which has the apogee in it. Now in the quadrant BE the force retards the moon and the moon is in apogee. Therefore the excentricity increases. In this situation therefore of the apogee, such is the average result of a complete revolution of the moon. Here again also if the perigee and apogee change places, so will also the character of all the partial influences, arc for arc. But the quadrant AD will now preponderate instead of D E, so that under this double reversal of conditions the result will be identical. Lastly, if the line of apsides be in A E, BD, it may be shown in like manner that the excentricity will diminish on the average of a revolution.

(691.) Thus it appears, that in varying the excentricity, precisely as in moving the line of apsides, the direct effect of the tangential force conspires with that of the normal, and tends to increase the extent of the deviations to and fro on either side of a mean value which the varying situation of the sun with respect to the line of apsides gives rise to, having for their period of restoration a synodical revolution of the sun and apse. Supposing the sun and apsis to start together, the sun of course will outrun the apsis (whose period is nine years), and in the lapse of about (+32) part of a year will have gained on it 90°, during all which interval the apse will have been in the quadrant A E of our figure, and the excentricity continually decreasing. The decrease will then cease, but the excentricity itself will be a minimum, the sun being now at right angles to the line of apsides. Thence it will increase to a maximum when the sun has gained another 90°, and again attained the line of apsides, and so on alternately. The actual effect on the numerical value

of the lunar excentricity is very considerable, the greatest and least excentricities being in the ratio of 3 to 2.*

(692.) The motion of the apsides of the lunar orbit may be illustrated by a very pretty mechanical experiment, which is otherwise instructive in giving an idea of the mode in which orbitual motion is carried on under the action of central forces variable according to the situation of the revolving body. Let a leaden weight be suspended by a brass or iron wire to a hook in the under side of a firm beam, so as to allow of its free motion on all sides of the vertical, and so that when in a state of rest it shall just clear the floor of the room, or a table placed ten or twelve feet beneath the hook. The point of support should be well secured from wagging to and fro by the oscillation of the weight, which should be sufficient to keep the wire as tightly stretched as it will bear, with the certainty of not breaking. Now, let a very small motion be communicated to the weight, not by merely withdrawing it from the vertical and letting it fall, but by giving it a slight impulse sideways. It will be seen to describe a regular ellipse about the point of rest as its center. If the weight be heavy, and carry attached to it a pencil, whose point lies exactly in the direction of the string, the ellipse may be transferred to paper lightly stretched and gently pressed against it. In these circumstances, the situation of the major and minor axes of the ellipse will remain for a long time very nearly the same, though the resistance of the air and the stiffness of the wire will gradually diminish its dimensions and excentricity. But if the impulse communicated to the weight be considerable, so as to carry it out to a great angle (15° or 20° from the vertical), this permanence of situation of the ellipse will no longer subsist. Its axis will be seen to shift its position at every revolution. of the weight, advancing in the same direction with the weight's motion, by an uniform and regular progression, which at length will entirely reverse its situation, bringing the direction of the longest excursions to coincide with that

Airy, Gravitation, p. 106.

in which the shortest were previously made; and so on, round the whole circle; and, in a word, imitating to the eye, very completely, the motion of the apsides of the moon's orbit.

(693.) Now, if we inquire into the cause of this progression of the apsides, it will not be difficult of detection. When a weight is suspended by a wire, and drawn aside from the vertical, it is urged to the lowest point (or rather in a direction at every instant perpendicular to the wire) by a force which varies as the sine of the deviation of the wire from the perpendicular. Now, the sines of very small arcs are nearly in the proportion of the arcs themselves; and the more nearly, as the arcs are smaller. If, therefore, the deviations from the vertical be so small that we may neglect the curvature of the spherical surface in which the weight moves, and regard the curve described as coincident with its projection on a horizontal plane, it will be then moving under the same circumstances as if it were a revolving body attracted to a center by a force varying directly as the distance; and, in this case, the curve described would be an ellipse, having its centre of attraction not in the focus, but in the center, and the apsides of this ellipse would remain fixed. But if the excursions of the weight from the vertical be considerable, the force urging it towards the center will deviate in its law from the simple ratio of the distances; being as the sine, while the distances are as the arc. the sine, though it continues to increase as the arc increases, yet does not increase so fast. So soon as the arc has any sensible extent, the sine begins to fall somewhat short of the magnitude which an exact numerical proportionality would require; and therefore the force urging the weight towards its center or point of rest at great distances falls, in like proportion, somewhat short of that which would keep the body in its precise elliptic orbit. It will no longer, therefore, have, at those greater distances, the same command over the weight, in proportion to its speed, which would enable it to

• Newton, Princip. i. 47.

Now

B

deflect it from its rectilinear tangential course into an ellipse. The true path which it describes will be less curved in the remoter parts than is consistent with the elliptic figure, as in the annexed cut; and, therefore, it will not so soon have its motion brought to be again at right angles to the radius. It will require a longer continued action of the central force to do this; and before it is accomplished, more than a quadrant of its revolution must be passed over in angular motion round the center. But this is only stating at length, and in a more circuitous manner, that fact which is more briefly and summarily expressed by saying that the apsides of its orbit are progressive. Nothing beyond a familiar illustration is of course intended in what is above said. The case is not an exact parallel with that of the lunar orbit, the disturbing force being simply radial, whereas in the lunar orbit a transversal force is also concerned, and even were it otherwise, only a confused and indistinct view of apsidal motion can be obtained from this kind of consideration of the curvature of the disturbed path. If we would obtain a clear one, the two foci of the instantaneous ellipse must be found from the laws of elliptic motion performed under the influence of a force directly as the distance, and the radial disturbing force being decomposed into its tangential and normal components, the momentary influence of either in altering their positions and consequently the directions and lengths of the axis of the ellipse must be ascertained. The student will find it neither a difficult nor an uninstructive exercise to work out the case from these principles, which we cannot afford the space to do.

(694.) The theory of the motion of the planetary apsides and the variation of their excentricities is in one point of view much more simple, but in another much more complicated than that of the lunar. It is simpler, because owing to the exceeding minuteness of the changes operated in the course of a single revolution, the angular position of the bodies with respect to the line of apsides is very little altered by the