Finding the orbits from the focus given

PROPOSITION 45 PROBLEM 31

Find the motion of the apsides in orbits approaching very near to circles

This problem is solved arithmetically by reducing the orbit, which a body revolving in a movable ellipsis (as in Cor. 2 and 3 of the above Prop.) describes in an immovable plane, to the figure of the orbit whose apsides are required. And then seeking the apsides of the orbit which that immovable plane.

This problem is solved arithmetically by reducing the orbit, which a body revolving in a movable ellipsis (as in Cor. 2 and 3 of the above Prop.) describes in an immovable plane, to the figure of the orbit whose apsides are required.

And then seeking the apsides of the orbit which that body describes in an immovable plane. But orbits acquire the same figure. if the centripetal forces with which they are described, compared between themselves, are made proportional at equal altitudes. Let the point V be the highest apsis, and write T for the greatest altitude CV, A for any other altitude CP or Cp, and X for the difference of the altitudes CV - CP; and the force with which a body moves in an ellipsis revolving about its focus C (as in Cor. 2), and which in Cor. 2 was as …, that is as … by substituting T - X for A; will become as …

In like manner any other centripetal force is to be reduced to a fraction whose denominator is A³, and the numerators are to be made analogous by collating together the homologous terms. This will be made plainer by Examples.

Example 1. Let us suppose the centripetal force to be uniform, and therefore as

… or, writing T - X for A in the numerator, as

….

Then collating together the correspondent terms of the numerators, that is, those that consist of given quantities, with those of given quantities, and those of quantities not given with those of quantities not given, it will become RGG - RFF + TFF to T³ as - FFX to 3TTX + 3TXX - X³, or as - FF to - 3TT + 3TX - XX.

Since the orbit is supposed extremely near to a circle, let it coincide with a circle; and because in that case R and T become equal, and X is infinitely diminished, the last ratios will be, as RGG to T², so - FF to - 3TT, or as GG to TT, so FF to 3TT; and again, as GG to FF, so TT to 3TT, that is, as 1 to 3; and therefore G is to F, that is, the angle VCp to the angle VCP, as 1 to ..

Therefore since the body, in an immovable ellipsis, in descending from the upper to the lower apsis, describes an angle, if I may so speak, of 180 deg., the other body in a movable ellipsis, and therefore in the immovable orbit we are treating of, will in its descent from the upper to the lower apsis, describe an angle VCp of … deg.

This comes to pass by reason of the likeness of this orbit which a body acted upon by an uniform centripetal force describes, and of that orbit which a body performing its circuits in a revolving ellipsis will describe in a quiescent plane. By this collation of the terms, these orbits are made similar; not universally, but then only when they approach very near to a circular figure.

A body, therefore revolving with an uniform centripetal force in an orbit nearly circular, will always describe an angle of .. deg., or 103 deg., 55 m., 23 sec., at the centre; moving from the upper apsis to the lower apsis when it has once described that angle, and thence returning to the upper apsis when it has described that angle again; and so on in infinitum.

Exam. 2

Suppose the centripetal force to be as any power of the altitude A, as, for example, An-3 3, or … where n - 3 and n signify any indices of powers whatever, whether integers or fractions, rational or surd, affirmative or negative.

That numerator An or … being reduced to an indeterminate series by my method of converging series, will become ….. &c.

Conferring these terms with the terms of the other numerator RGG - RFF + TFF - FFX, it becomes as RGG - RFF + TFF to Tn, so - FF to … &c.

Taking the last ratios where the orbits approach to circles, it becomes as RGG to Tn, so - FF to -nTn-1, or as GG to Tn-1, so FF to nTn-1.

Again, GG to FF, so Tn-1 to nTn-1, that is, as 1 to n; and therefore G is to F, that is the angle VCp to the angle VCP, as 1 to …

Therefore since the angle VCP, described in the descent of the body from the upper apsis to the lower apsis in an ellipsis, is of 180 deg., the angle VCp, described in the descent of the body from the upper apsis to the lower apsis in an orbit nearly circular which a body describes with a centripetal force proportional to the power An-3, will be equal to an angle of … deg.

This angle being repeated, the body will return from the lower to the upper apsis, and so on in infinitum. As if the centripetal force be as the distance of the body from the centre, that is, as A, or … n will be equal to 4, and … equal to 2.

Therefore the angle between the upper and the lower apsis will be equal to … deg., or 90 deg.

Therefore the body having performed a fourth part of one revolution, will arrive at the lower apsis, and having performed another fourth part, will arrive at the upper apsis, and so on by turns in infinitum.

This appears also from Prop. X. For a body acted on by this centripetal force will revolve in an immovable ellipsis, whose centre is the centre of force. If the centripetal force is reciprocally as the distance, that is, directly as … or … n will be equal to 2.

Therefore the angle between the upper and lower apsis will be … deg., or 127 deg., 16 min., 45 sec.

Therefore a body revolving with such a force, will by a perpetual repetition of this angle, move alternately from the upper to the lower and from the lower to the upper apsis for ever.

So, also, if the centripetal force be reciprocally as the biquadrate root of the eleventh power of the altitude, that is, reciprocally as … and, therefore, directly as … or as … n will be equal to ¼, and … deg. will be equal to 360 deg.

Therefore, the body parting from the upper apsis, and from thence perpetually descending, will arrive at the lower apsis when it has completed one entire revolution; and thence ascending perpetually, when it has completed another entire revolution, it will arrive again at the upper apsis; and so alternately for ever.

Exam. 3

Taking m and n for any indices of the powers of the altitude, and b and c for any given numbers, suppose the centripetal force to be as … that is, as … or (by the method of converging series above-mentioned) as …. &c.

and comparing the terms of the numerators, there will arise RGG - RFF + TFF to bTm + cTn as - FF to - mbTm-1 - ncTn-1 + &c.

Taking the last ratios that arise when the orbits come to a circular form, there will come forth GG to bTm-1 + cTn-1 as FF to mbTm-1 + ncTn-1; and again, GG to FF as bTm-1 + cTn-1 to mbTn-1 + ncTn-1. This proportion, by expressing the greatest altitude CV or T arithmetically by unity, becomes, GG to FF as b + c to mb + nc, and therefore as 1 to …

Whence G becomes to F, that is, the angle VCp to the angle VCP, as 1 to …

Therefore, since the angle VCP between the upper and the lower apsis, in an immovable ellipsis, is of 180 deg., the angle VCp between the same apsides in an orbit which a body describes with a centripetal force, that is, as … will be equal to an angle of … deg.

By the same reasoning, if the centripetal force be as … the angle between the apsides will be found equal to …..

After the same manner the Problem is solved in more difficult cases. The quantity to which the centripetal force is proportional must always be resolved into a converging series whose denominator is A³.

Then the given part of the numerator arising from that operation is to be supposed in the same ratio to that part of it which is not given, as the given part of this numerator RGG - RFF + TFF - FFX is to that part of the same numerator which is not given. And taking away the superfluous quantities, and writing unity for T, the proportion of G to F is obtained.

Corollary 1

Hence if the centripetal force be as any power of the altitude, that power may be found from the motion of the apsides; and so contrariwise.

That is, if the whole angular motion, with which the body returns to the same apsis, be to the angular motion of one revolution, or 360 deg., as any number as m to another as n, and the altitude called A; the force will be as the power … of the altitude A; the index of which power is …

This appears by the second example. Hence it is plain that the force in its recess from the centre cannot decrease in a greater than a triplicate ratio of the altitude. A body revolving with such a force and parting from the apsis, if it once begins to descend, can never arrive at the lower apsis or least altitude, but will descend to the centre, describing the curve line treated of in Cor. 3, Prop. XLI.

But if it should, at its parting from the lower apsis, begin to ascend never so little, it will ascend in infinitum, and never come to the upper apsis; but will describe the curve line spoken of in the same Cor., and Cor. 6; Prop. XLIV. So that where the force in its recess from the centre decreases in a greater than a triplicate ratio of the altitude, the body at its parting from the apsis, will either descend to the centre, or ascend in infinitum, according as it descends or ascends at the beginning of its motion.

But if the force in its recess from the centre either decreases in a less than a triplicate ratio of the altitude, or increases in any ratio of the altitude whatsoever, the body will never descend to the centre, but will at some time arrive at the lower apsis; and, on the contrary, if the body alternately ascending and descending from one apsis to another never comes to the centre, then either the force increases in the recess from the centre, or it decreases in a less than a triplicate ratio of the altitude.

The sooner the body returns from one apsis to another, the farther is the ratio of the forces from the triplicate ratio. As if the body should return to and from the upper apsis by an alternate descent and ascent in 8 revolutions, or in 4, or 2, or 1½; that is, if m should be to n as 8, or 4, or 2, or 1½ to 1, and therefore … be … or … or … or … then the force will be as or … or … or … that is, it will be reciprocally as … or … or … or …

If the body after each revolution returns to the same apsis, and the apsis remains unmoved, then m will be to n as 1 to 1, and therefore … will be equal to A-2, or …

Therefore, the decrease of the forces will be in a duplicate ratio of the altitude; as was demonstrated above.

If the body in three fourth parts, or two thirds, or one third, or one fourth part of an entire revolution, return to the same apsis; m will be to n as ¾ or ⅔ or ⅓ or ¼ to 1, and therefore … is equal to … or … or … or

Therefore the force is either reciprocally as … or … or directly as A6 or A13.

Lastly, if the body in its progress from the upper apsis to the same upper apsis again, goes over one entire revolution and three deg. more, and therefore that apsis in each revolution of the body moves three deg. in consequentia; then m will be to n as 363 deg. to 360 deg. or as 121 to 120, and therefore … will be equal to …

Therefore the centripetal force will be reciprocally as … or reciprocally as … very nearly.

Therefore the centripetal force decreases in a ratio something greater than the duplicate; but approaching 59¾ times nearer to the duplicate than the triplicate.

Corollary 2

Hence also if a body, urged by a centripetal force which is reciprocally as the square of the altitude, revolves in an ellipsis whose focus is in the centre of the forces; and a new and foreign force should be added to or subducted from this centripetal force, the motion of the apsides arising from that foreign force may (by the third Example) be known; and so on the contrary. As if the force with which the body revolves in the ellipsis be as …

and the foreign force subducted as cA, and therefore the remaining force as

…

then (by the third Example) b will be equal to 1. m equal to 1, and n equal to 4; and therefore the angle of revolution be tween the apsides is equal to 180 … deg.

Suppose that foreign force to be 357.45 parts less than the other force with which the body revolves in the ellipsis; that is, c to be 100 35745 {\displaystyle \scriptstyle {\frac {100}{35745}}}; A or T being equal to 1; and then 180

will be 180

or 180.7623, that is, 180 deg., 45 min., 44 sec. Therefore the body, parting from the upper apsis, will arrive at the lower apsis with an angular motion of 180 deg., 45 min., 44 sec.

This angular motion being repeated, will return to the upper apsis; and therefore the upper apsis in each revolution will go forward 1 deg., 31 min., 28 sec. The apsis of the moon is about twice as swift.

So much for the motion of bodies in orbits whose planes pass through the centre of force. It now remains to determine those motions in eccentrical planes. For those authors who treat of the motion of heavy bodies used to consider the ascent and descent of such bodies, not only in a perpendicular direction, but at all degrees of obliquity upon any given planes; and for the same reason we are to consider in this place the motions of bodies tending to centres by means of any forces whatsoever, when those bodies move in eccentrical planes.

These planes are supposed to be perfectly smooth and polished, so as not to retard the motion of the bodies in the least.

Moreover, in these demonstrations, instead of the planes upon which those bodies roll or slide, and which are therefore tangent planes to the bodies, I shall use planes parallel to them, in which the centres of the bodies move, and by that motion describe orbits. And by the same method I afterwards determine the motions of bodies performed in curve superficies.