tion of another, then three revolutions of the one would ex- CHAP. VI. actly correspond to two of the other, and every second conjunction of the two would take place in the same points of the orbits; and the orbits, not being circular, the portions of them on each side of the line of conjunctions cannot be symmetrical, unless the longer axes of the two orbits are in the same line, and the conjunctions also taking place on that line. Here, then, is a case showing that the disturbing force may constantly differ in amount on each side of the line of conjunctions, and, of course, could never compensate each other, and a permanent derangement of these two planets would be the result.

tem.

Hence, we perceive, that, to preserve the solar system, it Stability of is necessary that the orbits should be circles, or their times the solar sysof revolution incommensurable; but we do not pretend to say that the converse of this is true: we do say, however, that no natural cause of destruction has thus far been found.

(196.) The times of the planetary revolutions are incommensurable; but, nevertheless, there are instances that approach commensurability, and, in consequence, approach a derangement in motion, which, when followed out, produce very long periods of inequality, called secular variation. The most remarkable of these, and one which very much perplexed the astronomers of the last century, is known by the term of "the great inequality" of Jupiter and Saturn.

The great

It had long been remarked by astronomers that, on comparing together ancient with modern observations of Jupiter inequalities of Jupiter and Saturn, their mean motions could not be uniform." The and Saturn. period of Saturn appeared to have been increased throughout the whole of the seventeenth century, and that of Jupiter shortened. Saturn was constantly lagging behind its calculated place, and Jupiter was as constantly in advance of his. On the other hand, in the eighteenth century, a process precisely the reverse was going on.

The amount of retardations and accelerations, corresponding to one, two, or three revolutions were not very great; but, as they went on accumulating, material differences, at length, existed between the observed and calculated places of both

The perplexity given to the philo.

sophers.

CHAP. VI. these planets; and, as such differences could not then be ac counted for, they excited a high degree of attention, and formed the subject of prize problems of several philosophical societies.

Laplace

solved mystery.

For a long time these astonishing facts baffled every enthe deavor to account for them, and some were on the point of declaring the doctrine of universal gravity overthrown; but, at length, the immortal Laplace came forward, and showed the cause of these discrepancies to be in the near commensurability of the mean motions of Jupiter and Saturn; which cause we now endeavor to bring to the mind of the reader in a clear and emphatic manner.

The revo

piter and Sa

(197.) The orbits of both Jupiter and Saturn are elliptical, and their perihelion points have different longitudes, and, therefore, their different points of conjunction are at different distances from each other, and no line* of conjunction cuts the two orbits into two equal or symmetrical parts; hence, the inequalities of a single synodical revolution will not destroy each other; and, to bring about an equality of perturbations, requires a certain period or succession of conjunctions, as we are about to explain.

Five revolutions of Jupiter require 21663 days, and two lutions of Ju- of Saturn, 21518 days. So that, in a period of two revoluturn compar- tions of Saturn (about sixty of our years), after any conjunction of these two planets, they will be in conjunction again not many degrees from where the former took place.

ed.

Their syno

tion deter

To determine definitely where the third mean conjunction dical revolu- will take place, we compute the synodical revolution of these mined. two planets by dividing the circumference of the circle in seconds (1296000) by the difference of the mean daily motion of the planets in seconds (178".6),† and the quotient is 7253.4 days; three times this period is 21760 days. In this period Jupiter performs five revolutions and 8° 6' over; Saturn makes two revolutions and 8° 6' over; showing that the line

*Line of conjunction, an imaginary line drawn from the sun through the two planets when in conjunction.

+ See problem of the two couriers, Robinson's Algebra.

of conjunction advances 8° 6' in longitude during the period CHAP. VI, of 21760 days.

In the year 1800, the longitude of Jupiter's perihelion point was 11° 8', and that of Saturn 89° 9'; the inclination of the greater axis of the orbits, therefore, was 78° 1'.

Let AB. (Fig. 42) represent the major axis of Saturn's The series orbit, and ab that of Jupiter; the two are placed at an angle of 78°.*

of conjuno tions ex

plained.

plained.

Suppose any conjunction to take place in any part of the orbits, as at JS (the line JS we call the line of conjunc- Line of contion); in 7253.4 days afterward another conjunction will take junction explace. In this interval, however, Saturn will describe about 243° in its orbit, at a mean rate, and Jupiter will describe one revolution and about 243° over, and it will take place as represented in the figure, at P Q (STB being the direction of the motion). The next conjunction will be 243° from PQ, or at RT. From RT the next conjunction will be at si, 8°6′ in advance of JS, and thus the conjunction JS (so to speak) will gradually advance along on the orbit from S to T.

But, as we perceive, by inspecting the figure, there is a

*We have very much exaggerated the eccentricities of these ellipses, for the purpose of magnifying the principle under consideration.

CHAP. VI. certain portion of the orbits, between S and T, where the two Certain planets would come nearer together in their conjunction, than conjunctions they do at conjunctions generally, and, of course, while any

bring the pla

nets

most others.

able

nearer one of the three conjunctions is passing through that portion together than of the orbits - Jupiter disturbs Saturn, and Saturn reacts on Jupiter more powerfully than at other conjunctions; and this is the cause of "the great inequality of Jupiter and Saturn." The period of (198.) To obtain the period of this inequality, we comthis remarkine pute the time requisite for one of these lines of conjunction quality com- to make a third of a revolution, that is, divide 120° by 8° 6', puted, and and we shall find a quotient of 1424, showing the period to be tion confirm- 142 times 21760 days, or nearly 883 years; which would be ed by obser- the actual period, provided the elements of the orbits remained unchanged during that time. But in so long a period the relative position of the perigee points will undergo considerable variation; which causes the period to lengthen to about 918 years.

the computa

vation.

An expla nation of the principle that led to the discovery of Neptune.

P

Fig. 43.
S

The maximum amount of this inequality, for the longitude of Saturn, is 49', and for Jupiter 21', always opposite in effect, on the principle of action and reaction.

(199.) The last great achievement of the powers of mind in the solar system, was the discovery of the new planet Neptune, by Leverrier and Adams analyzing the inequalities of the motion

of Uranus. To give a rude explanation of the possibility of this problem, we present Figure 43. Let S be the sun, and the regular curve the orbit of Uranus, as corresponding to all known perturbations; but at a it departs from its computed track and runs out in the protuberance a cb. This indicated that some attracting body must be somewhere in the direction

How com

putations

SP, although no such body was ever seen or known to exist. CHAP. VI. The next time the planet comes round into the same portions of its orbit,* suppose the center of the protuberance to have changed to the line SQ. This would indicate that the unknown and unseen body was now in the line SQ, and that could be since the former observations it had changed positions by the made for the angle PSQ; and, by this angle, and the time of its description, something like a guess could be made of the time of its planet. revolution.

With the approximate time of revolution, and the help of Kepler's third law, its corresponding distance from the sun can be known. With the distance of the unseen body, and the amount that Uranus is drawn from its orbit by it, we can approximate to its mass.

Thus, we perceive, that it is possible to know much about an existing planet, although so distant as never to be seen. But the body that disturbed the motion of Uranus has been seen, and is called Neptune.

revolution of

an

unseen

CHAPTER VII.

ABERRATION, NUTATION, AND PRECESSION OF THE EQUINOXES.

CHAP. VII.

Dr. Bradley's obser

on

stars for the

purpose of

(200.) ABOUT the year 1725 Dr. Bradley, of the Greenwich observatory, commenced a very rigid course of observations on the fixed stars, with the hope of detecting their vations parallax. These observations disclosed the fact, that all the the fixed stars which come to the upper meridian near midnight, have an increase of longitude of about 20"; while those opposite, finding their near the meridian of the sun, have a decrease of longitude of Unexpected 20"; thus making an annual displacement of 40′′. These results. observations were continued for several years, and found to be the same at the same time each year; and, what was most

* Leverrier and Adams had not the advantage of a complete revolution of Uranus.

parallax.