自然哲学的数学原理-22

ferent planes ; and the force LM, acting in the direction of the line PTsituate in the plane of the orbit PAB, will have the same effect as before;neither will it draw the body P from the plane of its orbit. But the otherforce NM acting in the direction of a line parallel to ST (and which, therefore, when the body S is without the line of the nodes is inclined to theplane of the orbit PAB), besides the perturbation of the motion just nowspoken of as to longitude, introduces another perturbation also as to latitude,attracting the body P out of the plane of its orbit. And this perturbation,in any given situation of the bodies P and T to each other, will be as thegenerating force MN ; and therefore becomes least when the force MN isleast, that is (as was just now shewn), where the attraction SN is not nrirbgreater nor much less than the attraction SK. Q.E.D.SK-C. XL] OF NATURAL PHILOSOPHY. 205COR. 1. Hence it may be easily collected, that if several less bodies P8, R, &c.; revolve about a very great body T, the motion of the innermostrevolving body P will be least disturbed by the attractions of the others.when the great body is as well attracted and agitated by the rest (according to the ratio of the accelerative forces) as the rest are by each othermutually.COR. 2. In a system of three bodies, T, P, S, if the accelerative attractions of any two of them towards a third be to each other reciprocally as thesquares of the distances, the body P, by the radius PT, will describe its areaabout the body T swifter near the conjunction A and the opposition B than itwill near the quadratures C arid D. For every force with which the body Pis acted on and the body T is not, and which does not act in the direction ofthe line PT, does either accelerate or retard the description of the area,according as it is directed, whether in consequentia or in cwtecedentia.Such is the force NM. This force in the passage of the body P frcm Cto A is directed in consequentia to its motion, and therefore acceleratesit; then as far as D in atttecedentia, and retards the motion; then in, consequentia as far as B ; and lastly in antecedentia as it moves from B to C.COR. 3. And from the same reasoning it appears that the body P ccBterisparibuSj moves more swiftly in the conjunction and opposition than in thequadratures.COR. 4. The orbit of the body P, cc&teris paribus, is more curve at thequadratures than at the conjunction and opposition. For the swifterbodies move, the less they deflect from a rectilinear path. And besides theforce KL, or NM, at the conjunction and opposition, is contrary to theforce with which the body T attracts the body P, and therefore diminishesthat force ; but the body P will deflect the less from a rectilinear path theless it is impelled towards the body T.COR. 5. Hence the body P, cceteris paribus, goes farther from the bodyT at the quadratures than at the conjunction and opposition. This is said,E C_ LBhowever, supposing no regard had to the motion of eccentricity. For ifthe orbit of the body P be eccentrical, its eccentricity (as will be shewnpresently by Cor. 9) will be greatest when the apsides are in the syzygies;and thence it may sometimes come to pass that the body P. in itsnear approach to the farther apsis, may go farther from the body T at thesyzygies than at the quadratures.COR. 6. Because the centripetal force of the central body T, by which206 THE MATHEMATICAL PRINCIPLES [BOOK. 1the body P is retained in its orbit, is increased at the quadratures by thoaddition caused by the force LM, and diminished at the syzygies by thesubduction caused by the force KL, and, because the force KL is greaterthan LM, it is more diminished than increased ; and, moreover, since thatcentripetal force (by Cor. 2, Prop. IV) is in a ratio compounded of the simple ratio of the radius TP directly, and the duplicate ratio of the periodical time inversely ;it is plain that this compounded ratio is diminished bythe action of the force KL ; and therefore that the periodical time, supposingthe radius of the orbit PT to remain the same, will be increased, and thatin the subduplicate of that ratio in which the centripetal force is diminished ; and, therefore, supposing this radius increased or diminished, the periodical time will be increased more or diminished less than in the sesquiplicateratio of this radius, by Cor. 6, Prop. IV. If that force of the centralbody should gradually decay, the body P being less and less attracted wouldgo farther and farther from the centre T ; and, on the contrary, if it wereincreased, it would draw nearer to it. Therefore if the action of the distantbody S, by which that force is diminished, were to increase and decreaseby turns, the radius TP will be also increased and diminshed by turns ;and the periodical time will be increased and diminished in a ratio compounded of the sesquiplicate ratio of the radius, and of the subduplicate oithat ratio in which the centripetal force of the central body T is diminished or increased, by the increase or decrease of the action of the distantbody S.COR. 7. It also follows, from what was before laid down, that the axisof the ellipsis described by the body P, or the line of the apsides, does asto its angular motion go forwards and backwards by turns, but more forwards than backwards, and by the excess of its direct motion is in thewhole carried forwards. For the force with which the body P is urged tothe body T at the quadratures, where the force MN vanishes, is compounded of the force LM and the centripetal force with which the body T attracts the body P. The first force LM, if the distance PT be increased, isincreased in nearly the same proportion with that distance, and the otherforce decreases in the duplicate ratio of the distance ; and therefore thesum of these two forces decreases in a less than the duplicate ratio of thedistance PT ;and therefore, by Cor. 1, Prop. XLV, will make the line ofthe apsides, or, which is the same thing, the upper apsis, to go backward.But at the conjunction and opposition the force with which the body P isurged towards the body T is the difference of the force KL, and of theforce with which the body T attracts the body P ; and that difference, because the force KL is very nearly increased in the ratio of the distancePT, decreases in more -than the duplicate ratio of the distance PT ; andtherefore, by Cor. 1, Prop. XLV, causes the line of the apsides to go forwards. In the places between the syzygies and the quadratures, the motionSEC. Xl.J OF NATURAL PHILOSOPHY. 207of the line of the apsides depends upon both < f these causes conjuncdy, sothat it either goes forwards or backwards in proportion to the excess olone of these causes above the other. Therefore since the force KL in thesyzygies is almost twice as great as the force LM in the quadratures, theexcess will be on the side of the force KL, and by consequence the line ofthe apsides will be carried forwards. The truth of this arid the foregoingIECorollary will be more easily understood by conceiving the system of thetwo bodies T and P to be surrounded on every side by several bodies S,S, S, dec., disposed about the orbit ESE. For by the actions of these bodies the action of the body T will be diminished on every side, and decreasein more than a duplicate ratio of the distance.COR. 8. IJut since the progress or regress of the apsides depends uponthe decrease of the centripetal force, that is, upon its being in a greater orless ratio than the duplicate ratio of the distance TP, in the passage ofthe body from the lower apsis to the upper ; and upon a like increase inits return to the lower apsis again ; and therefore becomes greatest wherethe proportion of the force at the upper apsis to the force at the lower apsis recedes farthest from the duplicate ratio of the distances inversely ;itis plain, that, when the apsides are in the syzygies, they will, by reason ofthe subducting force KL or NM LM, go forward more swiftly ; and inthe quadratures by the additional force LM go backward more slowly.Because the velocity of the progress or slowness of the regress is continuedfor a long time ;this inequality becomes exceedingly great.COR. 9. If a body is obliged, by a force reciprocally proportional to thesquare of its distance from any centre, to revolve in an ellipsis round thatcentre ; and afterwards in its descent from the upper apsis to the lowerapsis, that force by a perpetual accession of new force is increased in morethan a duplicate ratio of the diminished distance ;it is manifest that thebody, being impelled always towards the centre by the perpetual accessionof this new force, will incline more towards that centre than if it wereurged by that force alone which decreases in a duplicate ratio of the diminished distance, and therefore will describe an orbit interior to thatelliptical orbit, and at the lower apsis approaching nearer to the centrethan before. Therefore the orbit by the accession of this new force willbecome more eccentrical. If now, while the body is returning from thelower to the upper apsis, it should decrease by the same degrees by whichit increases before the body would return to its first distance; and thereTHE MATHEMATICAL PRINCIPLES [BOOK I.fore if the force decreases in a yet greater ratio, the body, being now lessattracted than before, will ascend to a still greater distance, and so the eccentricity of the orbit will be increased still more. Therefore if the ratioof the increase and decrease of the centripetal force be augmented eachrevolution, the eccentricity will be augmented also ; and, on the contrary,if that ratio decrease, it will be diminished.Now, therefore, in the system of the bodies T, P, S, when the apsides ofthe orbit FAB are in the quadratures, the ratio of that increase and decrease is least of all, and becomes greatest when the apsides are in thesyzygies. If the apsides are placed in the quadratures, the ratio near theapsides is less, and near the syzygies greater, than the duplicate ratio of thedistances : and from that Greater ratio arises a direct motion of the line of7 othe apsides, as was just now said. But if we consider the ratio of thewhole increase or decrease in the progress between the apsides, this is lessthan the duplicate ratio of the distances. The force in the lower is to theforce in the upper apsis in less than a duplicate ratio of the distance of theupper apsis from the focus of the ellipsis to the distance of the lower apsisfrom the same focus ; and, contrariwise, when the apsides are placed in thesyzygies, the force in the lower apsis is to the force in the upper apsis in agreater than a duplicate ratio of the distances. For the forces LM in thequadratures added to the forces of the body T compose forces in a less ratio; and the forces KL in the syzygies subducted from the forces of thebody T, leave the forces in a greater ratio. Therefore the ratio of thewhole increase and decrease in the passage between the apsides is least atthe quadratures and greatest at the syzygies ; and therefore in the passageof the apsides from the quadratures to the syzygies it is continually augmented, and increases the eccentricity of the ellipsis ; and in the passagefrom the syzygies to the quadratures it is perpetually decreasing, and diminishes the eccentricity.COR. 10. That we may give an account of the errors as to latitude, letus suppose the plane of the orbit EST to remain immovable; and fromthe cause of the errors above explained, it is manifest, that, of the twoforces NM, ML, which are the only and entire cause of them, the forceML acting always in the plane of the orbit PAB never disturbs the motions as to latitude ; and that the force NM, when the nodes are in thegyzygies, acting also in the same plane of the orbit, does not at that timeaffect those motions. But when the nodes are in the quadratures, it disturbs tliem very much, and, attracting the body P perpetually out of theplane of its orbit, it diminishes the inclination of the plane in the passageof the body from the quadratures to the syzygies, and again increases thesame in the passage from the syzygies to the quadratures. Hence itcomes to pass that when the body is in the syzygies, the inclination isthen least of all, and returns to the first magnitude nearly, when the bodySEC. XL] OF NATURAL PHILOSOPHY. 209arrives at the next node. But if the nodes are situate at the octants afterthe quadratures, that is, between C and A, D and B, it will appear, fromii C LEwnat was just now shewn, that in the passage of the body P from eithernode to the ninetieth degree from thence, the inclination of the plane isperpetually diminished ; then, in the passage through the next 45 degreesto the next quadrature, the inclination is increased ; and afterwards, again,in its passage through another 45 degrees to the next node, it is diminished. Therefore the inclination is more diminished than increased, andis therefore always less in the subsequent node than in the preceding one.And, by a like reasoning, the inclination is more increased than diminished when the nodes are in the other octants between A and D, B and C.The inclination, therefore, is the greatest of all when the nodes are in thesyzygies In their passage from the syzygies to the quadratures the inclination is diminished at each appulse of the body to the nodes : and becomes least of all when the nodes are in the quadratures, and the body inthe syzygies ; then it increases by the same degrees by which it decreasedbefore ; and, when the nodes come to the next syzygies, returns to itsformer magnitude.COR. 11. Because when the nodes are in the quadratures the body P isperpetually attracted from the plane of its orbit ; and because this attraction is made towards S in its passage from the node C through the conjunction A to the node D ; and to the contrary part in its passage from thenode D through the opposition B to the node C; it is manifest that, in itsmotion from the node C, the body recedes continually from the formerplane CD of its orbit till it comes to the next node; and therefore at thatnode, being now at its greatest distance from the first plane CD, it willpass through the plane of the orbit EST not in D, the other node of thatplane, but in a point that lies nearer to the body S, which therefore becomes a new place of the node in, antecedentia to its former place. And,by a like reasoning, the nodes will continue to recede in their passagefrom this node to the next. The nodes, therefore, when situate in thequadratures, recede perpetually ; and at the syzygies, where no perturbation can be produced in the motion as to latitude, are quiescent : in the intermediate places they partake of both conditions, and recede more slowly ;and, therefore, being always either retrograde or stationary, they will becarried backwards, or in atitecedentia, each revolution.COR. 12. All the errors described in these corrollaries arc a little greater14210 THE MATHEMATICAL PRINCIPLES BOOK Lat the conjunction of the bodies P, S, than at their opposition ; becausethe generating forces NM and ML are greater.COR. 13. And since the causes and proportions of the errors and variations mentioned in these Corollaries do not depend upon the magnitude ofthe body S, it follows that all things before demonstrated will happen, ifthe magnitude of the body S be imagined so great as that the system of thetwo bodies P and T may revolve about it. And from this increase of thebody S, and the consequent increase of its centripetal force, from which theerrors of the body P arise, it will follow that all these errors, at equal distances, will be greater in this case, than in the other where the body S revolves about the system of the bodies P and T.COR. 14. But since the forces NM, ML, when the body S is exceedinglydistant, are very nearly as the force SK and the ratio PT to ST conjunctly;that is, if both the distance PT, and the absolute force of the body8 be given, as ST 3reciprocally : and since those forces NM, ML are thecauses of all the errors and effects treated of in the foregoing Corollaries;it is manifest that all those effects, if the system of bodies T and P continue as before, and only the distance ST and the absolute force of the bodyS be changed, will be very nearly in a ratio compounded of the direct ratioof the absolute force of the body S, and the triplicate inverse ratio of thedistance ST. Hence if the system of bodies T and P revolve about a distant body S, those forces NM, ML, and their eifl ts, will be (by Cor. 2 and6, Prop IV) reciprocally in a duplicate ratio c/f the periodical time. Andthence, also, if the magnitude of the bodv S be proportional to its absoluteforce, those forces NM, ML, and their effects, will be directly as the cubeof the apparent diameter of the distant body S viewed from T, and so viceversa. For these ratios are the same as the compounded ratio above mentioned.COR. 15. And because if the orbits ESE and PAB, retaining their figure, proportions, and inclination to each other, should alter their magnitude ;arid the forces of the bodies S and T should either remain, or bechanged in any given ratio ; these forces (that is, the force of the body T,which obliges the body P to deflect from a rectilinear course into the orbitPAB, and the force of the body S, which causes the body P to deviate fromthat orbit) would act always in the same manner, and in the same proportion : it follows, that all the effects will be similar and proportional, aridthe times of those effects proportional also; that is, that all the linear errors will be as tne diameters of the orbits, the angular errors the same asbefore ; and the times of similar linear errors, or equal angular errors? asthe periodical times of the orbits.COR. 16. Therefore if the figures of the orbits and their inclination toeach other be given, and the magnitudes, forces, arid distances of the bodieshe any how changed, we may. from the errors and times of those errors inSEC. XI.] OF NATURAL PHILOSOPHY. 2 。。one case, collect very nearly the errors and times of the errors in any othercase. But this may be done more expeditiously by the following method.The forces NM; ML, other things remaining unaltered, are as the radiusTP ; and their periodical effects (by Cor. 2, Lein. X) are as the forces andthe square of the periodical time of the body P conjunctly. These are thelinear errors of the body P ; and hence the angular errors as they appearfrom the centre T (that is, the motion of the apsides and of the nodes, and allthe apparent errors as to longitude and latitude) are in each revolution ofthe body P as the square of the time of the revolution, very nearly. Letthese ratios be compounded with the ratios in Cor. 14, and in any systemof bodies T, P, S, where P revolves about T very near to it, and T revolves about S at a great distance, the angular errors of the body P, observed from the centre T, will be in each revolution of the body P as thesquare of the periodical time of the body P directly, and the square of theperiodical time of the body T inversely. And therefore the mean motionof the line of the apsides will be in a given ratio to the mean motion ofthe nodes ; and both those motions will be as the periodical time of thebody P directly, and the square of the periodical time of the body T inversely. The increase or diminution of the eccentricity and inclination ofthe orbit PAB makes no sensible variation in the motions of the apsides*and nodes, unless that inc/case or diminution be very great indeed.COR. 17. Sines the line LM becomes sometimes greater and sometimesless than the radius PT, let the mean quantity of the force LM be expressedE Csa --::-..::::::;by that radius PT ; and then that mean force will be to the mean forceSK or SN (which may be also expressed by ST) as the length PT to thelength ST. But the mean force SN or ST, by which the body T is retained in the orbit it describes about S, is to the force with which the body Pis retained in its orbit about T in a ratio compounded of the ratio of theradius ST to the radius PT, and the duplicate ratio of the periodical timeof the body P about T to the periodical time of the body T about S. And,ex cequo, the mean force LM is to the force by which the body P is retained in its orbit about T (or by which the same body P might revolve at thedistance PT in the same periodical time about any immovable point T) inthe same duplicate ratio of the periodical times. The periodical timestherefore being given, together with the distance PT, the mean force LMis also given ; and that force being given, there is given also the force MN,very nearly, by the analogy of the lines PT and MN.212 THE MATHEMATICAL PRINCIPLES [BoOK ICon. IS. By tlie same laws by which the body P revolves about thebody T, let us suppose many fluid bodies to move round T at equal distances from it; and to be so numerous, that they may all become contiguousto each other, so as to form a fluid annulus, or ring, of a round figure, andconcentrical to the body T; and the several parts of this annulus, performing their motions by the same law as the body P, will draw nearer to thebody T, and move swifter in the conjunction and opposition of themselvesand the body S, than in the quadratures. And the nodes of this annulus,or its intersections with the plane of the orbit of the body S or T, will restat the syzygies ; but out of the syzygies they will be carried backward, orin. antecedentia ; with the greatest swiftness in the quadratures, and moreslowly in other places. The inclination of this annulus also will vary, andits axis will oscillate each revolution, and when the revolution is completedwill return to its former situation, except only that it will be carried rounda little by the precession of the nodes.COR. 19. Suppose now the spherical body T, consisting of some matternot fluid, to be enlarged, and to extend its If on every side as far as thatannulus, and that a channel were cut all round its circumference containing water j and that this sphere revolves uniformly about its own axis inthe same periodical time. This water being accelerated and retarded byturns (as in the last Corollary), will be swifter at the syzygies, and slowerat the quadratures, than the surface of the globe, and so will ebb and flow inits channel after the manner of the sea. If the attraction of the body S weretaken away, the water would acquire no motion of flux and reflux by revolv-.ng round the quiescent centre of the globe. The case is the same of a globemoving uniformly forwards in a right line, and in the mean time revolvingabout its centre (by Cor. 5 of the Laws of Motion), and of a globe uniformly attracted from its rectilinear course (by Cor. 6, of the same Laws).But let the body S come to act upon it, and by its unequable attraction theA。ater will receive this new motion ;for there will be a stronger attractionupon that part of the water that is nearest to the body, and a weaker uponthat part which is more remote. And the force LM will attract the waterdownwards at the quadratures, and depress it as far as the syzygies ; and theforce KL will attract it upwards in the syzygies, and withhold its descent,and make it rise as far as the quadratures ; except only in so far as themotion of flux and reflux may be directed by the channel of the water, andbe a little retarded by friction.COR. 20. If, now, the annulus becomes hard, and the globe is diminished,the motion of flux and reflux will cease ; but the oscillating motion of theinclination and the praecession of the nodes will remain. Let the globehave the same axis with the annulus, and perform its revolutions in thesame times, and at its surface touch the annulus within, and adhere to it;then the globe partaking of the motion of the annulus, this whole comparesSEC. XI. OF NATURAL PHILOSOPHY. 213will oscillate, and the nodes will go backward, for the globe, as 。ve shallshew presently, is perfectly indifferent to the receiving of all impressions.The greatest angle of the inclination of the annulus single is when thenodes are in the syzygies. Thence in the progress of the nodes to thequadratures, it endeavours to diminish its inclination, and by that endeavour impresses a motion upon the whole globe. The globe retains thismotion impressed, till the annulus by a contrary endeavour destroys thatmotion, and impresses a new motion in a contrary direction. And by thismeans the greatest motion of the decreasing inclination happens when thenodes are in the quadratures; and the least angle of inclination in the octantsBafter the quadratures ; and, again, the greatest motion of roclination happenswhen the nodes are in the syzygies ; and the greatest angle of reclination inthe octants following. And the case is the same of a globe without this annulus, if it be a little higher or a little denser in the equatorial than in thepolar regions : for the excess of that matter in the regions near the equatorsupplies the place of the annulus. And though we should suppose the centripetal force of this globe to be any how increased, so that all its partswere to tend downwards, as the parts of our earth gravitate to the centre,yet the phenomena of this and the preceding Corollary would scarce be altered ; except that the places of the greatest and least height of the waterwill be different : for the water is now no longer sustained and kept in itsorbit by its centrifugal force, but by the channel in which it flows. And,besides, the force LM attracts the water downwards most in the quadratures, and the force KL or NM LM attracts it upwards most in thesyzygies. And these forces conjoined cease to attract the water downwards,and begin to attract it upwards in the octants before the syzygies ; andcease to attract the water upwards, and begin to attract the water down