lion, the line of apsides will regress if the disturbing force should act after the passage of the aphelion, but will progress if it should act at the mean distance or before the passage of the perihelion; while again, the eccentricity in each of these three cases will be increased by the disturbing force.

(22.) We have seen that if the disturbing force should act when the planet is at the perihelion, the effect is then thrown wholly upon the apsides, which rapidly progress; that at the mean distance the apsides also progress, (though with less rapidity, the effect now being thrown partly upon the apsides and partly on the eccentricity); but that if the disturbing force acts when the planet arrives at the aphelion, the effect is again thrown wholly upon the apsides, which, however, in this case regress. Hence it is obvious, that there must be some intermediate point of the orbit between the mean distance and the aphelion at which the disturbing force produces no effect on the position of the line of apsides. Similarly, it is manifest that there must exist some point between the aphelion and the subsequent point of mean distance, at which the line of apsides does not undergo any change of position from the action of the disturbing force. It may be found by a simple investigation, to which we shall presently allude more particularly, that the two points in question are the extremities F G of the ordinate passing through H the upper

B

H

A

G

focus of the ellipse. In fact, as the planet revolves from F to G through A, the line of apsides everywhere progresses from the action of the disturbing force, the amount of progression increasing from nothing at F until it attains its maximum at A, and subsequently diminishing until it vanishes again at G. Similarly, from a to F through B, the line of apsides everywhere regresses from the same cause, the amount of regression being greatest at A, and diminishing in either direction towards F and G.

(23.) We have seen that if, when the planet is revolving from the lower to the upper apse, the disturbing force act a little after the passage of the perihelion, at the mean distance, or a little before the passage of the aphelion, the eccentricity is in each case diminished; but that, on the other hand, when the planet is revolving from the upper to the lower apse, the eccentricity in each of the corresponding cases is increased by the action of the disturbing force. Generally it may be shewn, that from A to B through G the eccentricity is everywhere diminished by the action. of the disturbing force, the amount of diminution increasing from nothing at A until it attains its maximum at G, and subsequently diminishing until it vanishes at B; and on the other hand, that from B to A through F the eccentricity is everywhere increased by the action of the disturbing force, the amount of increase being greatest at F, and diminishing from that point towards A and B. Thus it appears, that when the variation in the position of the apsides is greatest, the variation of the eccentricity is least,

and vice versa.

(24.) If we suppose the disturbing force to act in the direction of the radius vector so as to diminish the central force at s, it will be found, in a similar manner, that the effect both upon the eccentricity and the apsides will now be precisely the reverse of that produced when the disturbing force acts inwards. In this case the eccentricity will increase from a to

B through G, and will diminish from B to A through F; and on the other hand, the apsides will regress from F to G through A, and will progress from G to F through B. The points where the variations of both elements attain their maximum values, and also those at which they severally vanish, will be the same as in the former case.

(25.) Let us now consider the effect of a small disturbing force acting in a direction perpendicular to the radius vector. In all such cases the eccentricity of the orbit is supposed to be so inconsiderable, that the disturbing force may be regarded as acting in the direction of the tangent, and consequently as tending wholly either to accelerate or retard the motion of the planet. Let us suppose, then, that it tends to increase the velocity of the planet, and first let it act when the planet is at the perihelion. The velocity being now increased, the central force will have less control over the planet, and the latter in consequence taking a wider sweep, will now recede, farther at the mean distance. It is manifest, also, since the tangential angle continually enlarges from the perihelion to the mean distance, that it will now open out to a greater extent than it formerly did. Now, the greater the maximum value of the tangential angle, the more eccentric is the orbit. Hence the effect of the disturbing force is to increase the mean distance, and also the eccentricity. It is manifest that the position of the line of apsides cannot suffer any alteration from the action of such a force.

(26.) Let us now suppose the disturbing force to act at the aphelion. Since the velocity is increased, the central force will be less effective in deflecting the motion of the planet, and the latter in consequence taking a wider circuit, will not approach so near the centre of force at the mean distance as it formerly did. Moreover it is manifest, since the diminution of the tangential angle continues from the aphelion to the mean distance, that when it attains its minimum value, it will be less acute than it formerly was. Hence the effect of the disturbing force in this case is, to increase the mean distance, and to diminish the eccentricity.

(27.) It is manifest that the conclusions above deduced are equally applicable if we suppose the disturbing force to act at a little distance on each side of the apse, whether the latter refer to the perihelion or the aphelion, for the circumstances which determine the path of the planet are then almost the same as if the disturbing force had acted exactly at the apse.

(28.) Let us suppose the planet to be revolving from the lower to the upper apse, and let the disturbing force act when it has arrived at the

T

E

E

T

B

B

mean distance E. At this point the deflection of motion in the undisturbed orbit exactly compensates for the angular displacement of the radius. vector, and the tangential angle in consequence remains for an instant invariable. The velocity of the planet, however, being now increased by the action of the disturbing force, the momentary deflection of motion will be diminished, and therefore the tangential angle will still continue to open out. Let E' be the point at which the momentary deflection of motion in the new orbit is equal to the cor

responding change of direction of the radius vector, and let E'T be a tangent to the orbit at that point. Then, since the angle S E'T' is greater than S E T, it follows that the eccentricity is increased by the disturbing force. It is evident, however, that the increase of that element is very small, for the planet is in the most favourable position for the retardation of its motion by the central force, and consequently the deflection of motion is speedily brought to an equality with the angular displacement of the radius vector. The position of the major axis of the new orbit will be determined by drawing A' B' parallel to E' T', whence it is evident that the line of apsides has progressed. It may be shewn, in a similar manner, that when the planet is revolving from the upper to the lower apse, the effect of the disturbing force at the mean distance is, to diminish the eccentricity, and to make the line of apsides regress. In every point of the orbit, the mean distance is obviously increased by the action of the disturbing force.

(29.) Since the eccentricity is increased at E and diminished at B by the action of the disturbing force, it is evident that there is some intermediate point at which the disturbing force produces no effect upon that element. For a similar reason, there must be some point between B and the following point of mean distance at which the eccentricity does not undergo any change from the action of the disturbing force. These neutral points are found by strict investigation to be FG, the extremities of the ordinate passing through the upper focus of the ellipse. (See the figure at page 590). (30.) Generally it may be shewn, that from F to G through A, the eccentricity is everywhere increased by the action of the disturbing force, the variation increasing from nothing at F until it attains its maximum at A, and subsequently diminishing by equal degrees until it vanishes at G. On the other hand, from G to F through B, the eccentricity is everywhere diminished by the action of the disturbing force, the variation being a maximum at B, from which point it diminishes towards F and G.

(31.) With respect to the line of apsides, it has been found to progress if the planet, while revolving from the perihelion to the aphelion, should be at the mean distance when the disturbing force acts, and to regress, if the planet, in the course of revolving from aphelion to perihelion, should have arrived at the corresponding point of the orbit; while, again. at either apse it does not undergo any change of position. Generally it may be shewn, that from A to B through G, the line of apsides progresses, and that from B to A through F, it regresses, the variation attaining its maximum values at F and G, and vanishing at A and B. Thus it appears, that in passing from a disturbing force acting in the direction of the radius vector, to one acting at right angles to that direction, an interchange takes place between the points of maxima and minima of the variations of the eccentricity and the apse.

(32.) If the disturbing force act in a direction contrary to that of the planet's motion so as to diminish the velocity, the effects produced upon the eccentricity and the line of apsides will be precisely the reverse of those abovementioned. The eccentricity will increase from A to B through G, and will diminish from B to A through F; while again the line of apsides will regress from F to G through A, and will progress from a to F through B. Moreover the points at which the variations attain their maximum values, and also those at which they vanish, will be the same as in the case wherein the disturbing force tends to increase the velocity of the planet.

(33.) The variation induced in the position of the line of apsides by a disturbing force of given intensity is greater, as the eccentricity of the orbit is less. The truth of this proposition will appear manifest on a very slight consideration of the subject. It is easily seen that a variation in the position of the line of apsides is tantamount to a variation of the tangential angle in any point of the orbit. Now the variation of the tangential angle is slower as the eccentricity is less, the difference between its least and greatest values continually diminishing, until at length, when the orbit becomes a circle, the difference vanishes altogether, and the tangential angle is constantly of the same magnitude. The same is obviously true if the difference refer to any two values of the tangential angle comprehended within the extreme values. Hence it follows that the displacement of the line of apsides which will be required in order to adapt the orbit to a given alteration in the magnitude of the tangential angle due to a disturbing force of given intensity at any point, will be greater as the eccentricity of the orbit is less.

(34.) Since the tangential angle varies to a greater extent as the disturbing force is more intense, we may therefore infer conversely, that in order to induce an alteration of given magnitude in the position of the line of apsides, the intensity of the disturbing force must be greater as the orbit is more eccentric.

(35.) Hitherto we have supposed the disturbing force to act for a short space of time and then to cease. If its action be constantly kept up as in every case of planetary perturbation, the alteration effected in any of the elements of the orbit during a given interval of time, may be ascertained by investigating the change for each successive instant, and then summing up the results. It is easy by means of the foregoing principles to determine the character of the effect produced in any such case, although its exact amount can only be ascertained by a process of computation based on the principles of the infinitesimal calculus.

(36.) Mr. Airy has shewn (Gravitation, notes to Arts. 50 and 65), that by a slight modification of the figure given by Newton in Prop. XVII. of the first book of the Principia, the effects produced on the eccentricity and the position of the line of apsides by a force acting either in the direction of the radius vector, or along the tangent of the orbit, may be clearly exhibited to the eye. Sir John Herschel has actually employed this mode of expounding the variations of the elements in question in his recently-published Outlines of Astronomy. The simplicity and elegance of the investigation will amply justify its insertion here.

T

T'

B'

(37.) First let us suppose the disturbing force to act in the direction of the radius vector, so as to increase the attractive force at s. We have seen that the direct effect of such a force is to diminish the tangential angle. Let it act at P, and let the tangent T T at that point be deflected in consequence, so as now to occupy the position x x'. Draw P H' so that the angle H' P X' shall be equal to s P X, the new value of the tangential angle. From P, set off PH'equal to PH. Bisect s H' in c'. Then is s c' the new eccentricity of the orbit, and A's H'B' the new position of the line of apsides. For, by the property of the ellipse, the two lines drawn from any given point of it to the foci make equal angles with the

H

H

tangent. Hence the upper focus of the new ellipse must be somewhere in the line P H'. Again, by another property of the ellipse, the sum of the same two lines is equal to the major axis. Hence s and H' will represent the foci of an ellipse whose tangential angle at P is equal to s P X, and whose major axis is equal to that of the undisturbed orbit. But by the principles of dynamics, the major axis of the ellipse is not altered by the disturbing force. Hence the truth of the proposition is manifest. By varying the position of P, and supposing the disturbing force as tending either to increase or diminish the attractive force at s, the different results referred to in (22), (23), and (24) may be very easily deduced.

(38.) Next let the disturbing force act in the direction of the tangent, so as to retard or accelerate the velocity of the planet. Let it be supposed to increase the velocity and let P be the point at which it acts. În

T

H

T

B

this case the tangential angle at P is not affected by the disturbing force, but the major axis of the ellipse is increased by its action. Produce PH so that H H' may represent the increment of the major axis occasioned by the disturbing force. Join s H', and bisect the line s H in c'. Then will s c' represent the new eccentricity of the orbit, and A'S H' B' the new position of the line of apsides. The truth of this proposition is so obvious as to render any formal demonstration of it superThe various theorems announced in (30), (31), and 32) are easily deducible from it.

པ་

fluous.

H'

II.

APPLICATION OF THE FOREGOING PRINCIPLES TO CERTAIN CASES OF ACTUAL

PERTURBATION.

(39). Let us suppose two comparatively small bodies to be revolving in circular orbits situate in the same plane, round a large central body s, and let the mean motion of the interior revolving body be almost exactly double the mean motion of the exterior one. Let us assume also, for facility of explanation, that the exterior body maintains a fixed position at P, while the interior

D

body performs an entire revolution around s. Join s P by a straight line, cutting the orbit of the interior body in A and D. Then DAP will represent the line of conjunction of the two bodies. Now in those cases of the solar system wherein the mean motion of one revolving body is almost exactly double the mean motion of the other, the effects produced by the mutual perturbation of the two bodies are sensible only near conjunction. Let us suppose that in one of such cases the disturbing influence of the exterior body first becomes sensible when the interior body has arrived at B, a position somewhat less advanced than the line of conjunction. It may be easily shewn that the disturb