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planets, and of the moon. 3. Disturbances in the inotions of the primary planets, from their actions on one another. 4. Disturbances in the motions of Jupiter's satellites from their mutual actions, with the general result from the theory of the planetary disturbances. 5. Attraction of spheres and splieroids. 6. Figure of the earth. 7. Precession of the equinoxes, variation of the diurnal rotation and of the obliquity of the ecliptic. 8. Physical explanation of the phenomena of the tides, and concluding remarks on the principle of universal gravitation.
We shall extract some instructive passages from the commencement of the second section :
When there are only two bodies that gravitate to one another, with forces inversely as the squares of their distances, it appears from the last section that they move in conic sections, and describe about their common centre of gravity, equal areas in equal times, that centre either remaining at rest, or moving uniformly in a right line. But if there are three bodies, the action of any one on the other two, changes the nature of their orbits, so that the determination of their motions becomes a problem of the greatest difficulty, distinguished by the name of THE PROBLEM OF THREE BODIES.
• The solution of this problem in its utmost generality, is not within the power of the mathematical sciences, as they now exist. Under certain limitations, however, and such as are quite consistent with the condition of the heavenly bodies, it admits of being resolved, These limitations are, that the force which one of the bodies exerts on the other two, is, either from the smallness of that body, or its great distance, very inconsiderable in respect of the forces which these two exert on one another.
• The force of this third body is called a disturbing force, and its effects in changing the places of the other two bodies are called the disturbances of the system.
• Though the small disturbing forces may be more than one, or though there be a great number of remote disturbing bodies, the computation of their combined effect arises readily from knowing the effect of one ; and therefore the problem of three bodies, under the conditions just stated, may be extended t any number
• Two very different methods have been applied to the solution of this problem The most perfect is that whics embraces all the effects of the disturbances at once, and by reducing the momentary changes into Auxionary or differential equations, proceeds, by the integration of these, to determine the whole change produced in any finite time, whether on the angular or the rectilineal distance of the bodies. This method gives all the inequalities at once, and as they mutually affect one another.
· The other method of solution is easier, and more elementary, but inuch less accurate. It supposes the orbit disturbed, to be nearly known, and proceeds to calculate each inequality by itself, independently of the rest. It cannot, therefore, be exact, and gives only a first approximation to the quantities sought: but being far simpler
than the other, it is much better suited to the elements of science, It is also the original method, and that which was first applied by Sir ISAAC NEWTON, to explain the irregularities of the moon's motion, The same has been followed and improved, by CALENDRINI, in his Commentary on the third Book of the Principia; by Frist in his Cosa mographia ; and by Vince in the second volume of his Astronomy.
• The other method was not invented till several years later, when it occurred nearly about the same time to the three first geometers of the age, CLAIRAUT, EULER, and D'Alembert. It was followed also by Mayer, and several others, but particularly by LAPLACE, who, in the Mecanique Celeste, has given a complete investigation of the inequalities both of the primary and secondary planets.
• I shall explain the resolution of the forces that is in some measure common to both methods ; and shall shew how their effects are to be estimated in some simple instances, going from thence to the enu. meration of the results. I begin with the moon's irregularities, as the easiest case of the problem.'
These he traces with considerable perspicuity, stating the most important propositions, and enumerating many curious particulars, especially those which tend to confirm the assumed theory of gravitation. We have room to specify only one of them. Clairaut after solving the problem which relates to the motion of the apsides of the lunar orbit, on comparing the result with observation, met with the same difficulty that Newton liad experienced, and * Found that his formula gave only half the true motion.
He therefore imagined that gravity is not inversely as the squares of the distances, but follows a more complicated law, such as can only be expressed by a formula of two terms. In seeking for the co efficient of the second term, he was obliged to carry his approximation farther than he had done before ; in consequence of which the co-efficient he sought for came out equal to nothing, and the motion of the apsides was found to be completely explained by the supposition that the force of gravity is inversely as the square of the distance.
Another striking confirmation, as well as application, of this universal theory, is given at p. 282, when our Author is treating of comets, and the way in which their orbits are atfected by the disturbing forces of the planets. He also presents a few obseryations on the improbability that any perceptible alteration in the motion of the planets, or indeed any sensible effect upon them, should be produced by comets. This subject, by the way, is treated in a very satisfactory manner by Delambre, in his quarto Astronomie, tome iii. p. 404-6, and in the Abrigé, p. 564. The latter work is frequently cited by the Professor.
After developing the principal effects of the disturbing forces of the planets upon the several parts of the solar system, he
terminates this portion of his investigations by the following instrụctive and interesting conclusion.
• One general result of these investigations is, that both in the system of primary and secondary planets, two elements of every orbit remuin secure against all disturbance; the mean distance and the mean motion, or, which is the same, the transverse axis of the orbit and the time of the planet's revolution. Another result is, that all the inequalities in the planetary motions are periodical, and observe such laws that each of them after a certain time runs through the same series of changes
• Every inequality is expressed by terms of the form A sin nt or A cob nt ; where is a constant co-efficient, anda a certain multiplier of t the time, so that nt is an arc of a circle which increases proportionally to the time. Now, though nt is thus capable of indefinite increase, since sin nt never can exceed the radius or 1, the maximum of the inequality is A. Accordingly, the value of the term A sin nt first increases from o to A, and then decreases from A to o; after which it becomes negative, extends to = -t, and passes from thence to o again, the period of all those changes depending on n the multiplier of t.
• If into the value of any of the inequalities, a term of the form, A tan nt, or of the form A X nt were to enter, the inequality so
sin nt expressed, would continually increase, and the order of the system might finally be displaced.
LA GRANG and La Place, in demonstrating that no such terms as these last can enter into the expression of the disturbances of the planets, made known one of the most important truths in phy. sical astronomy. They proved that the system is stable; that it does not involye any principle of destruction in itself, but is calculated to endure for ever, unless the action of an external power is introduced
• This accurate compensation of the inequalities of the planetary motions, depends on three conditions, belonging to the primitive and original constitution of the system.
'1. That the eccentricities of the orbits are all inconsiderable, or contained within very narrow limits.
• II hat the Planets all move in the same direction, as both primary and secondary do from west to east.
• Ill. That the planes of their orbits are but little inclined to one another,
• But for these three conditions, terms of the kind mentioned above, would come into the expression of the inequalities, which might therefore increase without limit.
· These three conditions do not necessarily arise out of the nature of motion or of gravitation, or from the action of any physical cause with which we are acquainted. Neither can they be considered as arising from chance ; for the probability is almost infinite to one, that without a cause particularly directed to that object, such a conformity
could not have arisen in the motions of 31 different bodies scattered over such a vast extent.
· The only explanation therefore that remains is, that all this is the work of intelligence and design, directing the original constitution of the system and expressing such motions on the parts as were calculated to give stability to the whole.'
This, as far as it goes, is excellent. But the principle of gra. vitation will enable us to take avother step, and that a very momentous one. It is demonstrable from this principle, not only that there existed originally a Designing Agent, but that the universal system requires his perpetual intervention.
This has been shown conclusively by many writers, but by none, perhaps, more indubitably than by Professor Vince in his « Obis servations on the Cause of Gravitation” which we reviewed some years ago.
• It seems reasonable (says Mr. Vince) to admit a Divine Agency at that point where all other means appear inadequate to produce the effect. And as mechanical operations, in whatever point of view they have been considered, do not appear sufficient to account for the preservation of the system (to say nothing of its formation), we ought to conclude, that the Deity, in his government, does not act by such instruments ; but that the whole is conducted by his more immediate agency, without the intervention of material causes.'
A mathematical writer in a celebrated northern journal, la boured hard to weaken this consolatory inference : but, bappily, he failed in the attempt by neglecting (whether from ignorance or intention we cannot say) to distinguish between motive and accelerating force.
There is much valuable matter in the remainder of these “ Outlines,” but we have not room to speak of more than a single topic, viz. the variation of the obliquity of the ecliptic. The position of the ecliptic is incessantly changing by reason of the action of the planets.
• The variations in the obliquity of the ecliptic, thus produced, are among the number of the secular inequalities which have long periods, and, after reaching a maximum, return in a contrary direction.
* As far back as observation goes, the obliquity of the ecliptic has been diminishing, and is doing so at present, by 52" in a century; it will not, however, always continue to diminish, but in the course of ages will again increase, oscillating backwards and forwards on each side of a mean, from which it never can depart far.
The secular variation of the obliquity was less in ancient times than it is at present; it is now near its maximum, and will begin to decrease in the 22d century of our era.
· La Grange has shewn, that the total change of the obliquity,
reckoning from that in 1700, must be less than 50 23'; Mem. Acad. de Berlin, 1782. p. 284. Also that the changes in the inclinations of the planetary orbits, are all periodical, and cannot carry the planes of those orbits beyond the limits of the zodiac, or 8° on either side of the ecliptic. By the retrogradations of the nodes of the ecliptic and the planetary orbits, the precession of the equinoxes is diminished by a small quantity, which is at present about 0' 281 annually. Ibid. p. 281. All this is quite independent of the figure of the earth, and would be the same though the earth were truly spherical.
These variations in the obliquity, with their limits and peculiarities, will become still more manifest to the student, on his applying the curious theorem given by Laplace for that purpose. Let i denote the number of years from 1750, to be regarded as negative before, and as positive after that epoch ; then will the obliquity be always nearly expressed in sexigesimal measures by the formula, 23°28'23." 05 - 1191•2184 [1 -- cos (t. 13•194645)]
- 3347•' 0496 sin. (t. 32.111575). It is interesting to observe how the sentiments of astronomers have vacillated on this subject. Copernicus and Kepler were both of opinion, not only that the obliquity varied, but that the variation had limits. The former assigned them between 23° 56' and 23' 281; the latter, between 26° 5' and 22° 20',-a most remarkable conjecture, considering the time in which it was advanced. Afterwards, in the seventeenth century, and at the beginning of the eighteenth, philosophers in general aimed to prove that the obliquity was constant. Thus, Professor Bernard, of Oxford, in a paper published A. D. 1684, in No. 163, of the Philosophical Transactions, endeavoured to prove there was no diminution: and Flamstead, by transmitting that paper to the Royal Society unaccompanied by any remark, seemed to concur in the opinion. Dom. Cassini, Labire, and even Lemonnier, so late as 1715, took the same side of the question.
In 1716, when M. de Louville presented to the French Academy of Sciences a paper in which he attempted to prove that the obliquity was actually diminishing, that paper was not adinitted into their memoirs, because all the astronomical Academicians thought differently from Louville. Malgré toutes • les raisons de M. de Louville (said Fontenelle, in the History
of the Academy for 1816) les autres astronomes de l'académie sont demeurés attachés à l'obliquité constante de l'ecliptique
de 230 29'.' The disquisition being thus excluded from the Paris Memoirs, was inserted in the “ Acta Eruditorum,” of Leipsic, for June 1719. Such, however, is now the state of physical astronomy, that if a person were to call in question the fact of the variations of the obliquity, he would be ex