Principia Mathematica (Motte Translation, 1848) — part 15 of 45

former T and L with the accelerative forces ST, SL, and let it be attract- ed again by them. The force ST (by Cor. 2, of the Laws of Motion) is resolved into the forces SD, DT; and the force SL into the forces SD and DL. Now the forces DT, DL, which are as their sum TL, and therefore as the accelerative forces wdth which the bodies T and L attract each other mutually, added to the forces of the bodies T and L, the first to the first, and the last to the last, compose forces proportional to the distances DT and DL as before, but only greater than those former forces ; and there- fore (by Cor. 1, Prop. X, and Cor. 1, and 8, Prop. lY) they will cause those bodies to describe ellipses as before, but with a swifter motion. The re- maining accelerative forces SD and DL,by the motive forces SD X Tand SD X L, which are as the bodies attracting those bodies equally and in the direction of the lines TI, LK parallel to DS,do not at all change their situ- ations with respect to one another, but cause them equally to approach to the line IK ; which must be imagined drawn through the middle of the body S, and perpendicular to the line DS. But that approach to the line

2Q0 THE MATHEMATICAL PRINCIPLES [BoOK I.

IK will be hindered by causing the system of the bodies T and L on one side, and the body S on the other, with proper velocities, to revolve round the common centre of gravity C. With such a motion the body S, because the sum of the motive forces SD X T and SD X L is proportional to the distance CS, tends to the centre C, will describe an ellipsis round the same centre C; and the point D, because thd^lines CS and CD are proportional, will describe a like ellipsis over against it. But the bodies T and L, at- tracted by the motive forces SD X T and SD X L, the first by the first, and the last by the last, equally and in the direction of the parallel lines Tl and LK, as was said before, will (by Cor. 5 and 6, of the Laws of Motion) continue to describe their ellipses round the movable centre D, as before. Q.E.L

Let there be added a fourth body Y, and, by the like reasoning, it will be demonstrated that this body and the point C will describe ellipses about the common centre of gravity B ; the motions of the bodies T, L, and S round the centres D and C remaining the same as before-; but accelerated. And by the same method one may add yet more bodies at pleasure. Q.E.I.

This would be the case, though the bodies T and L attract each other mutually with accelerative forces either greater or less than those with which they attract the other bodies in proportion to their distance. Let all the mutual accelerative attractions be to each other as the distances multiplied into the attracting bodies ; and from what has gone before it will easily be concluded that all the bodies will describe different ellipses with equal periodical times about their common centre of gravity B, in an immovable plane. Q.E.I.

PROPOSITION LXY. THEOREM XXY.

Bodies, tu hose forces decrease in a duplicate ratio of their distances from their centres, may Quove among themselves in ellipses ; and by radii drawn to the foci may describe areas proportional to the times very nearly.

In the last Proposition we demonstrated that case in which the motions will be performed exactly in ellipses. The more distant the law of the forces is from the law in that case, the more will the bodies disturb each other's motions ; neither is it possible that bodies attracting each other mutually according to the law supposed in this Proposition should move exactly in ellipses, unless by keepirg a certain proportion of distances from each other. However, in the following cases the orbits will not much dif- fer from ellipses.

Case L Imagine several lesser bodies to revolve about some very great one at different distances from it, and suppose absolute forces tending to every one of the bodies proportional to each. And because (by Cor. 4, of the Laws) the common centre of gravity of them all is either at rest, or

Sec. XL] of natural philosophy. 201

moves uniformly forward in a right line, suppose the lesser bodies so small that the great body may be never at a sensible distance from that centre ; and then the great body will/ without any sensible error, be either at rest, or move uniformly forward in a right line ; and the lesser will revolve about that great one in ellipses, and by radii drawn thereto will describe areas proportional to the times ; if we except the errors that may be intro- duced by the receding of the great body from the common centre of gravity, or by the mutual actions of the lesser bodies upon each other. But the lesser bodies may be so far diminished, as that this recess and the mutual actions of the bodies on each other may become less than any assignable; and therefore so as that the orbits may become ellipses, and the areas an- swer to the times, without any error that is not less than any assignable.

aE.o.

Case 2. Let us imagine a system of lesser bodies revolving about a very great one in the manner just described, or any other system of two bodies revolving about each other to be moving uniformly forward in a right line, and in the mean time to be impelled sideways by the force of another vastly greater body situate at a great distance. And because the equal accelerative forces with which the bodies are impelled in parallel directions do not change the situation of the bodies with respect to each other, but only oblige the whole system to change its place while the parts still retain their motions among themselves, it is manifest that no change in those motions of the attracted bodies can arise from their attractions towards the greater, unless by the inequality of the accelerative attractions, or by the inclinations of the lines towards each other, in whose directions the attractions are made. Suppose, therefore, all the accelerative attractions made towards the great body to be among themselves as the squares of the distances reciprocally ; and then, by increasing the distance of the great body till the dilFerences of tTie right lines drawn from that to the others in respect of their length, and the inclinations of those lines to each other, be less than any given, the mo- tions of the parts of the system will continue without errors that are not less than any given. And because, by the small distance of those parts from each other, the whole system is attracted as if it were but one body, it will therefore be moved by this attraction as if it were one body ; that is, its centre of gravity will describe about the great body one of the conic sec- tions (that is, a parabola or hyperbola when the attraction is but languid, and an ellipsis when it is more vigorous) ; and by radii drawn thereto, it will describe areas proportional to the times, without any errors but those which arise from the distances of the parts, which are by the supposition exceedingly small, and may be diminished at pleasure. Q.E.O.

By a like reasoning one may proceed to more compounded cases in in- finihim.

Cor. 1. In the second Case, the nearer the very great body approaches to

202 THE MATHEMATICAL PRINCIPLES [BoOK I.

the system of two or more revolving bodies, tlie greater ■will the pertur- bation be of the motions of the parts of the system among themselves ; be- cause the inclinations of the lines drawn from that great body to those parts become greater ; and the inequality of the proportion is also greater.

Cor. 2. But the perturbation will be greatest of all, if we suppose the accelerative attractions of the parts of the system towards the greatest body of all are not to each other reciprocally as the squares of the distances from that great body ; especially if the inequality of this proportion be greater than the inequality of the proportion of the distances from the great body. For if the accelerative force^ acting in parallel directions and equally, causes no perturbation in the motions of the parts of the system, it must of course, when it acts unequally, cause a perturbation some- where, which will be greater or less as the inequality is greater or less. The excess of the greater impulses acting upon some bodies, and not acting upon others, must necessarily change their situation among themselves. And this perturbation, added to the perturbation arising from the inequality and inclination of the lines, makes the whole perturbation greater.

Cor. 3. Hence if the parts of this system move in ellipses or circles without any remarkable perturbation, it is manifest that, if they are at all impelled by accelerative forces tending to any other bodies, the impulse is very weak, or else is impressed very near equally and in parallel directions upon all of them.

PROPOSITION LXYI. THEOREM XXVI.

If three bodies whose forces decrease ifi a duplicate ratio of the distances attract each other mutually ; and the accelerative attractions of any tiDO toioards the third he between themselves reciprocally as the squares of the distances ; and the tioo least revolve about the greatest ; I say, that the interior of the tioo revolving bodies will, by radii drawn to the innermost and greatest, describe round that body areas onore propor- tional to the times, and a figure more approaching to that of an ellip- sis having its focus in the point of co7icourse of the radii, if that great body be agitated by those attractions, than it would do if that great body iDcre not attracted at all by the lesser, but remained at rest ; or than it iDoidd if that great body were very much onore or very much less attracted, or very much more or very much less agitated, by the attractions. This appears plainly enough from the demonstration of the second

Corollary of the foregoing Proposition ; but it may be made out after

this manner by a way of reasoning more distinct and more universally

convincing.

Case 1. Let the lesser bodies P and S revolve in the same plane about

the greatest body T, the body P describing the interior orbit PAB, and S

Sec. XL] of natural philosophy. 203

the exterior orbit ESE. Let SK be the mean distance of the bodies P and S ; and let the accelerative attraction of the body P towards S, at that mean distance, be expressed by that line SK. Make SL to SK as the

'E C L

-^-M

square of SK to the square of SP, and SL will be the accelerative attrac- tion of the body P towards S at any distance SP. Join PT, and draw LM parallel to it meeting ST in M ; and the attraction SL will be resolv- ed (by Cor. 2, of the Laws of Motion) into the attractions SM, LM. And s« the body P will be urged with a threefold accelerative force. One of these forces tends towards T, and arises from the mutual attraction of the bodies T and P. By this force alone the body P would describe round the body T, by the radius PT, areas proportional to the times, and an ellipsis whose focus is in the centre of the body T ; and this it would do whether the body T remained unmoved, or whether it were agitated by that attraction. This appears from Prop. XI, and Cor. 2 and 3 of Theor. XXL The other force is that of the attraction LM, which, because it tends from P to T, will be superadded to and coincide with the former force ; and cause the areas to be still proportional to the times, by Cor. 3, Theor. XXI. But because it is not reciprocally proportional to the square of the distance PT, it will compose, when added to_ the former, a force varying from that proportion ; which variation wdll be the greater by how much the proportion of this force to the former is greater, cceter is paribus. Therefore, since by Prop. XI, and by Cor. 2, Theor. XXI, the force with which the ellipsis is described about the focus T ought to be directed to that focus, and to be reciprocally proportional to the square of the distance PT, that compounded force varying from that proportion will make the orbit PAB vary from the figure of an ellipsis that has its focus in the point T ; and so much the more by how much the variation from that proportion is greater ; and by consequence by how much the proportion of the second force LM to the first force is greater, cceteris paribus. But now the third force SM, attracting the body P in a direction parallel to ST, composes with the other forces a new force which is no longer directed from P to T : and which varies so much more from this direction by how much the proportion of this third force to the other forces is greater, ccBteris paribus ; and therefore causes the body P to describe, by the radius TP, areas no longer proportional to the times ; and therefore makes the variation from that proportionality so much greater by how much the proportion of this force to the others is greater. But this third force will increase the variation of the orbit PAB from the

204 THE MATHEMATICAL PRINCIPLES [BoOK I.

elliptical figure before-mentioned upon two accounts ; first because that force is not directed from P to T ; and, secondly, because it is not recipro- cally proportional to the square of the distance PT. These things being premised, it is manifest that the areas are then most nearly proportional to the times, when that third force is the least possible, the rest preserving their former quantity , and that the orbit PAB does then approach nearest to the elliptical figure aboye-mentioned, when both the second and third, but especially the third force, is the least possible; the first force remain- ing in its former quantity.

Let the accelerative attraction of the body T towards S be expressed by the line SN ; then if the accelerative attractions SM and SN were equal, these, attracting the bodies T and P equally and in parallel directions would not at all change their situation with respect to each other. The mo- tions of the bodies between themselves would be the same in that case as if those attractions did not act at all, by Cor. 6, of the Laws of Motion. And^ by a like reasoning, if the attraction SN is less than the attraction SM, it will take away out of the attraction SM the part SN, so that there will re- main only the part (of the attraction) MN to disturb the proportionality of the areas and times, and the elliptical figure of the orbit. And in like manner if the attraction SN be greater than the attraction SM, the pertur- bation of the orbit and proportion will be produced by the difference MN alone. After this manner the attraction SN reduces always the attraction SM to the attraction MN, the first and second attractions remaining per- fectly unchanged ; and therefore the areas and times come then nearest to proportionality, and the orbit PAB to the above-mentioned elliptical figure, when the attraction MN is either none, or the least that is possible ; that is, when the accelerative attractions of the bodies P and T approach as near as possible to equality ; that is, when the attraction SN is neither none at all, nor less than the least of all the attractions SM, but is, as it were, a mean between the greatest and least of all those attractions SM, that is, not much greater nor much less than the attraction SK. Q..E.D.

Case 2. Let now the lesser bodies P, S, revolve about a greater T in dif- ferent planes ; and the force LM, acting in the direction of the line PT situate 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 other force NM acting in the direction of a line parallel to ST (and which, there- fore, when the body S is without the line of the nodes is inclined to the plane of the orbit PAB), besides the perturbation of the motion just now spoken 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 the generating force MN ; and therefore becomes least when the force MN is least, that is (as was just now shewn), where the attraction SN is not much greater nor much less than the attraction SK, Q.E.D.

Sec. XL] of natural philosophy. 205

• Cor. 1. Hence it may be easily collected, that if several less bodies P, S, R, &C.J revolve about a very great body T, the motion of the innermost revolving 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 (accord- ing to the ratio of the accelerative forces) as the rest are by each other mutually.

Cor. 2. In a system of three bodies, T, P, S, if the accelerative attrac- tions of any two of them towards a third be to each other reciprocally as tbe squares of the distances, the body P, by the radius PT, will describe its area iabout the body T swifter near the conjunction A and the opposition B than it will near the quadratures C and D. For every force with which the body P is acted on and the body T is not, and which does not act in the direction of the line PT, does either accelerate or retard the description of the area, according as it is directed, whether in consequentia or in antecedentia. Such is the force NM. This force in the passage of the body P from C to A is directed in consequentia to its motion, and therefore accelerates it; then as far as D in antecedentia, and retards the motion; then in con- sequentia 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 coiteris jparihus^ moves more swiftly in the conjunction and opposition than in the quadratures.

Cor. 4. The orbit of the body P, cceteris paribus, is more curve at the quadratures than at the conjunction and opposition. For the swifter bodies move, the less they deflect from a rectilinear path. And besides the force KL, or NM, at the conjunction and opposition, is contrary to the force with which the body T attracts the body P, and therefore diminishes that force ; but the body P will deflect the less from a rectilinear path the less it is impelled towards the body T.

Cor. 5. Hence the body P, cceteris paribus, goes farther from the body

T at the quadratures than at the conjunction and opposition. This is said,

/E C I,

P/TT::^ \

sd— ---

D

however, supposing no regard had to the motion of eccentricity. For if the orbit of the body P be eccentrical, its eccentricity (as will be shewn presently by Cor. 9) will be greatest when the apsides are in the syzy- gies ; and thence it may sometimes come to pass that the body P, in its near approach to the farther apsis, may go farther from the body T at the Byzygies than at the quadratures.

CoR, 6. Because the centripetal force of the central body T, by which

206 THE MATHEMATICAL PRINCIPLES [BoOK I.

the body P is retained in its orbit, is increased at tbe quadratures by the addition caused by the force LM, and diminished at the syzygies by the subduction caused by the force KL, and, because the force KL is greater than LM, it is more diminished than increased ; and, moreover, since that centripetal force (by Cor. 2, Prop. lY) is in a ratio compounded of the sim- ple ratio of the radius TP directly, and the duplicate ratio of the periodi- cal time inversely ; it is plain that this compounded ratio is diminished by the action of the force KL ; and therefore that the periodical time, supposing the radius of the orbit PT to remain the same, will be increased, and that in the subduplicate of that ratio in which the centripetal force is diminish- ed ; and, therefore, supposing this radius increased or diminished, the peri- odical time will be increased more or diminished less than in the sesquipli- cate ratio of this radius, by Cor. 6, Prop. IV. If that force of the central body should gradually decay, the body P being less and less attracted would go farther and farther from the centre T ; and, on the contrary, if it were increased, it would draw nearer to it. Therefore if the action of the distant body S, by which that force is diminished, were to increase and decrease by turns, the radius TP will be also increased and diminshed by turns ; and the periodical time will be increased and diminished in a ratio com- pounded of the sesquiplicate ratio of the radius, and of the subduplicate of that ratio in which the centripetal force of the central body T is dimin- ished or increased, by the increase or decrease of the action of the distant body S.

Cor. 7. It also follows, from what was before laid down, that the axis of the ellipsis described by the body P, or the line of the apsides, does as to its angular motion go forwards and backwards by turns, but more for- wards than backwards, and by the excess of its direct motion is in the whole carried forwards. For the force with which the body P is urged to the body T at the quadratures, where the force MN vanishes, is compound- ed of the force LM and the centripetal force with which the body T at- tracts the body P. The first force LM, if the distance PT be increased, is increased in nearly the same proportion with that distance, and the other force decreases in the duplicate ratio of the distance ; and therefore the sum of these two forces decreases in a less than the duplicate ratio of the distance PT ; and therefore, by Cor. 1, Prop. XLY, will make the line of the 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 is urged towards the body T is the difference of the force KL, and of the force with which the body T attracts the body P ; and that difference, be- cause the force KL is very nearly increased in the ratio of the distance PT, decreases in more than the duplicate ratio of the distance PT ; and therefore, by Cor. 1, Prop. XLY, causes the line of the apsides to go for- wards. In the places between the syzygies and the quadratures, the motion

Sec. XL] op natural philosophy. 207

of the line of the apsides depends upon both of these causes conjunctly, so that it either goes forwards or backwards in proportion to the excess of one of these causes above the other. Therefore since the force KL in the syzygies is almost twice as great as the force LM in the quadratures, the excess will be on the side of the force KL, and by consequence the line of the apsides will be carried forwards. The truth of this and the foregoing

fE C Ii

D

Corollary will be more easily understood by conceiving the system of the two bodies T and P to be surrounded on- every side by several bodies S, S, S, (fee, disposed about the orbit ESE. For by the actions of these bo- dies the action of the body T will be diminished on every side, and decrease in more than a duplicate ratio of the distance.

Cor. S. But since the progress or regress of the apsides depends upon the decrease of the centripetal force, that is, upon its being in a greater or less ratio than the duplicate ratio of the distance TP, in the passage of the body from the lower apsis to the upper ; and upon a like increase in its return to the lower apsis again ; and therefore becomes greatest where the proportion of the force at the upper apsis to the force at the lower ap- sis recedes farthest from the duplicate ratio of the distances inversely ; it is plain, that, when the apsides are in the syzygies, they will, by reason of the subducting force KL or NM — LM, go forward more swiftly ; and in the quadratures by the additional force LM go backward more slowly. Because the velocity of the progress or slowness of the regress is continued for a long time ; this inequality becomes exceedingly great.

Cor. 9. If a body is obliged, by a force reciprocally proportional to the square of its distance from any centre, to revolve in an ellipsis round that centre ; and afterwards in its descent from the upper apsis to the lower apsis, that force by a perpetual accession of new force is increased in more than a duplicate ratio of the diminished distance ; it is manifest that the body, being impelled always towards the centre by the perpetual accession of this new force, will incline more towards that centre than if it were urged by that force alone which decreases in a duplicate ratio of the di- minished distance, and therefore will describe an orbit interior to that elliptical orbit, and at the lower apsis approaching nearer to the centre than before. Therefore the orbit by the accession of this new force will become more eccentrical. If now, while the body is returning from the lower to the upper apsis, it should decrease by the same degrees by which it increases before the body would return to its first distance ; and there-

208 THE MATHEMATICAL PRINCIPLES [BoOK I.

fore if the force decreases in a yet greater ratio, the body, being now less attracted than before, will ascend to a still greater distance, and so the ec- centricity of the orbit will be increased still more. Therefore if the ratio of the increase and decrease of the centripetal force be augmented each revolution, 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 of the orbit PAB are in the quadratures, the ratio of that increase and de- crease is least of all, and becomes greatest when the apsides are in the syzygies. If the apsides are placed in the quadratures, the ratio near the apsides is less, and near the syzygies greater, than the duplicate ratio of the distances ; and from that greater ratio arises a direct motion of the line of the apsides, as was just now said. But if we consider the ratio of the whole increase or decrease in the- progress between the apsides, this is less than the duplicate ratio of the distances. The force in the lower is to the force in the upper apsis in less than a duplicate ratio of the distance of the upper apsis from the focus of the ellipsis to the distance of the lower apsis from the same focus ; and, contrariwise, when the apsides are placed in the syzygies, the force in the lower apsis is to the force in the upper apsis in a greater than a duplicate ratio of the distances. For the forces LM in the quadratures added to the forces of the body T compose forces in a less ra- tio ; and the forces KL in the syzygies subducted from the forces of the body T, leave the forces in a greater ratio. Therefore the ratio of the whole increase and decrease in the passage between the apsides' is least at the quadratures and greatest at the syzygies ; and therefore in the passage of the apsides from the quadratures to the syzygies it is continually aug- mented, and increases the eccentricity of the ellipsis ; and in the passage from the syzygies to the quadrat;ares it is perpetually decreasing, and di- minishes the eccentricity.

Cor. 10. That we may give an account of the errors as to latitude, let us suppose the plane of the orbit EST to remain immovable ; and from the cause of the errors above explained, it is manifest, that, of the two forces NM, ML, which are the only and entire cause of them, the force ML acting always in the plane of the orbit PAB never disturbs the mo- tions as to latitude ; and that the force NM, when the nodes are in the syzygies, acting also in the same plane of the orbit, does not at that time affect those motions. But when the nodes are in the quadratures, it dis- turbs them very much, and, attracting the body P perpetually out of the plane of its orbit, it diminishes the inclination of the plane in the passage of the body from the quadratures to the syzygies, and again increases the same in the passage from the syzygies to the quadratures. Hence it comes to pass that when the body is in the syzygies, the inclination is then least of all; and returns to the first magnitude nearly, when the body

Sec. XL] of natural ppiilosophy. 209

arrives at the next node. But if the nodes are situate at the octants after the quadratures; that is, between C and A, D and B, it will appear, from

/B C L

-M

D

what was just now shewn, that in the passage of the body P from either node to the ninetieth degree from thence, the inclination of the plane is perpetually diminished ; then, in the passage through the next 45 degrees to the next quadrature, the inclination is increased ; and afterwards, again, in its passage through another 45 degrees to the next node, it is dimin- ished. Therefore the inclination is more diminished than increased, and is therefore always less in the subsequent node than in the preceding one. And, by a like reasoning, the inclination is more increased than diminish- ed 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 the syzygies. In their passage from the syzygies to the quadratures the incli- nation is diminished at each appulse of the body to the nodes ; and be- comes least of all when the nodes are in the quadratures, and the body in the syzygies ; then it increases by the same degrees by which it decreased before ; and, when the nodes come to the next syzygies, returns to its former magnitude.

Cor. 11. Because when the nodes are in the quadratures the body P is perpetually attracted from the plane of its orbit ; and because this attrac- tion is made towards S in its passage from the node C through the con- junction A to the node D ; and to the contrary part in its passage from the node D through the opposition B to the node C; it is manifest that, in its motion from the node C, the body recedes continually from the former plane CD of its orbit till it comes to the next node; and therefore at that node, being now at its greatest distance from the first plane CD, it will pass through the plane of the orbit EST not in D, the other node of that plane, but in a point that lies nearer to the body S, which therefore be~ comes a new place of the node i?i antecedentia to its former place. And, by a like reasoning, the nodes will continue to recede in their passage from this node to the next. The nodes, therefore, when situate in the quadratures, recede perpetually ; and at the syzygies, where no perturba- tion can be produced in the motion as to latitude, are quiescent : in the in- termediate places they partake of both conditions, and recede more slowly :, and, therefore, being always either retrograde or stationary, they will be carried backwards, or in antecedentia, each revolution.

CoR. 12. All the errors described in these corrollaries arc a little greateir

14

210 THE MATHEMATICAL PRINCIPLES [BoOK I.

at the conjunction of the bodies P, S, than at their opposition ; because the generating forces NM and ML are greater.

CoR. 13. And since the causes and proportions of the errors and varia- tions mentioned in these Corollaries do not depend upon the magnitude of the body 8, it follows that all things before demonstrated will happen, if the magnitude of the body S be imagined so great as that the system of the two bodies P and T may revolve about it. And from this increase of the 'body S, and the consequent increase of its centripetal force, from which the errors of the body P arise, it will follow that all these errors, at equal dis- tances, will be greater in this case, than in the other where the body S re- volves about the system of the bodies P and 1'.

Cor. 14. But since the forces NM, ML, when the body S is exceedingly distant, are very nearly as the force SK and the ratio PT to ST con- junctly ; that is, if both the distance Pl^, and the absolute force of the body S be given, as ST^ reciprocally ; and since those forces NM, ML are the causes 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 con- tinue as before, and only the distance ST and the absolute force of the body S be changed, will be very nearly in a ratio compounded of the direct ratio of the absolute force of the body S, and the triplicate inverse ratio of the distance ST. Hence if the system of bodies T and P revolve about a dis- tant body S, those forces NM, ML, and their effects, will be (by Cor. 2 and 6, Prop IT) reciprocally in a duplicate ratio of the periodical time. And thence, also, if the magnitude of the body S be proportional to its absolute force, those forces NM, ML, and their effects, will be directly as the cube of the apparent diameter of the distant body S viewed from T, and so vice versa. For these ratios are the same as the compounded ratio above men- tioned.

Cor. 15. And because if the orbits ESE and PAB, retaining their fig- ure, proportions, and inclination to each other, should alter their magni- tude ; and the forces of the bodies S and T should either remain, or be changed 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 orbit PAB, and the force of the body S, which causes the body P to deviate from that orbit) would act always in the same manner, and in the same propor- tion ; it follows, that all the effects will be similar and proportional, and the times of those effects proportional also ; that is, that all the linear er- rors will be as the diameters of the orbits, the angular errors the same as before ; and the times of similar linear errors, or equal angular errors, as :the periodical times of the orbits.

Cor. 16. Therefore if the figures of the orbits and their inclination to

€ach other be given, and the magnitudes, forces, and distances of the b(*dies be any how changed, we may, from the errors and times of those errors in

Sec. XL] of natural philosophy. 211