5n't-2nt + 5 € - 2 € + 5° 34′ 8′′-t x 58".58 = 0,

and the value of t resulting from this equation is nearly 190, which, subtracted from 1750, leaves 1560 to represent the epoch at which Saturn's retardation, during the year, was the greatest, and equal to

About the year 1560, then, the observations must necessarily have shewn Saturn, in his greatest retardation, and (see p. 347.) Jupiter in his greatest acceleration. Or, with reference solely to their great inequality, the true motions of the above two planets differed most from their mean about the time of Tycho Brahe's observations; an epoch remarkable for the revival of Astronomical Science.

The explanation of Saturn's retardation, according to the law and principle of Gravity, was first given by Laplace in the Memoirs of the Academy of Paris for the years 1785, 1786. His researches, however, go beyond that explanation; and are extended to comprehend the complete theory of Jupiter and Saturn, or, what is in fact, a general solution of the Problem of the Three Bodies.

The solution of the great inequality of Jupiter and Saturn, on the principles of Physical Astronomy, marks, with considerable precision, the progress which that science has made since its rise. Its great founder, as we have remarked, (see p. 326.) noted no peculiarity in the theory of Jupiter and Saturn: nothing, which either strongly confirmed, or which seemed to form an exception to, his system. Yet, as we have seen, there are, in the motions of the above two planets, remarkable inequalities: which, for a long time were considered as anomalous, and which, as long as they were so considered, formed an exception amongst the results from the law of Gravity. They, however, most forcibly illustrated the truth of that law when they were proved not to be anomalous, and had been reduced to the class of other inequalities that arose from planetary perturbation.

The minute and very gradual variations and long periods of those inequalities that are the subject of the present discussion, have occasioned the practical difficulty of detecting them by observation. The difficulty of detecting their mathematical cause, has arisen from its lying concealed, as it were, amongst insignificant terms.

The cause, which in the theory of Jupiter and Saturn, gives importance to certain of these terms may operate in other cases. If the mean motions of any two planets are nearly commensurable, such planets are subject to inequalities of a very long period. But there is no rule, short of actual trial, for ascertaining whether any two planets are under the predicaments that Jupiter and Saturn are. We must examine the Table of Mean Motions, and, on making trials, we shall find (see Astronomy, p. 283.) that five times Mercury's period is nearly equal to twice Venus's, or, which is the same thing, twice the mean motion of Mercury (n) is nearly equal to five times that of Venus (n');

Now, some of the terms, in that part of the expression for 42 (see p. 333.) which involves the squares of the eccentricities, depend on the angle 3 nt - 5n't: and (see p. 333.) the integration introduces into the value of the divisor,

a2

5 n') - n2 = (2n - 5n) (4 n

,

5 n'),

which by reason of the factor 2 n − 5 n', becomes very small : and accordingly, in the theory of Mercury disturbed by Venus, it is necessary to attend to the inequality dependent on the angle

In the same manner, since the mean motion of Mercury is nearly equal (see Astronomy, p. 283.) to four times that of the Earth, we must attend, in the theory of Mercury disturbed by the Earth, to the inequality dependent on the angle 2nt-4n't in

rdr

the terms of the expression for that involve the squares of

a2

,

the eccentricities: for, here, the divisor introduced by integration is (2 n - 4 n")2 - n2, which contains the factors n - 4 n", and 3 n - 4n", the first of which is very small.

The terms just spoken of, involve the squares of the eccentricities and form part of the value of : but, if n-4n" be very

ror
a2

small, it will be necessary (see pp. 335, &c.) to attend to the terms in the expression for do which involve the cubes of the eccentricities, and depend on the anglent -4n"t: for, in such terms, a divisor (n - 4 n") is admitted.

The explanation of the alternate retardation and acceleration of Saturn and Jupiter affords, it has been said, a kind of practical proof of the progress which Physical Astronomy has made since the time of Newton: it shews also after what manner the latter science is superior to Plane Astronomy, and is capable of benefiting it. For although we may, by the aid of empirical equations and observations, determine inequalities of short periods, yet it seems impossible, by like means, to determine inequalities so protracted in their periods as those we have been discussing.

Observation is unequal to the task of disengaging them: they would always, without the aid of theory, appear so blended with the mean motions as either gradually to accelerate or retard them. Modern observations, for instance, compared with each other, and with antient observations, would, as Halley found it, make Saturn to appear retarded and Jupiter accelerated: or, modern observations, compared with each other, might, as Lambert found it to be the case, make Jupiter appear retarded and Saturn accelerated. The Tables of these two planets, before the causes, laws and quantities of their accelerations and retardations were ascertained, were erroneous to the amount of twenty-two minutes. Laplace's equations reduced the errors within two minutes; and the Tables are now exact to within a quarter of a minute: this is one of the practically good effects of theory: and in this, as in similar instances, it no longer goes hand in hand with observation, but advances before and serves it as a guide *.

By the theorem of p. 328. it follows, that if Saturn were subject to a really secular retardation from the action of Jupiter, Jupiter would suffer an acceleration equally secular from Saturn, and in the proportion of m' √a': -ma. Now the inequalities, as we have seen in p. 348. are not secular but periodical: their periods, however, are so long, that the inequalities are almost accurately in the above proportion: they would be less accurately so, were their periods shorter: still, however, as the fact is, not very inaccurate, were the periods very much short

ened.

Hence, if we should have investigated the acceleration produced, during a considerable period, in one planet's motion by the action of another, we might at once find the corresponding retardation produced, during the same period, in the motion of the disturbing body, by merely multiplying the first found acceleration by

ma

:

or, this last result might be used as a test of the truth

m'a' of the retardation computed by a direct process. For instance, the action of the Earth on Venus causes an inequality dependent on the angle 3n't 2n't: (n", n' denoting the mean motions of the Earth and Venus): the period of which, accordingly, (see p. 235, and Astron. p. 283.) is nearly four years, the inequality, expressed by its two parts†, is

* There is something curious in the history of the theory of Jupiter and Saturn. First, its peculiar phenomena were unnoticed by the great founder of Physical Astronomy: next, when noted and examined they seemed to impair his system; lastly, they have served, when explained and accounted for, most strongly to confirm it.

† This inequality might be expressed by a single term by means of the process of p. 344.

for the corresponding inequality in the Earth's motion caused by the action of Venus: which is, very nearly, an accurate result, since the two coefficients, deduced by a direct process, are - 1".08, + 3".6.

In like manner the action of the Earth on Venus produces an inequality dependent on the angle 5 n" t-3n't, and the period of which, accordingly, is about eight years. The inequality expressed by one term, is

1".5.sin. (5 n" t-3n't + 5€" - 3 € + 20° 54′ 28′′), and, if this denote a retardation, the coefficient of the corresponding acceleration in the Earth's motion produced by the action of Venus, is

which is very nearly the value of the coefficient resulting from the direct process.

If Venus then accelerate the Earth by a particular inequality for a certain period, the Earth will retard Venus by a like inequality and for the same period: the coefficient of the inequality will be different, the argument the same. This we know certainly by the mathematical process. But the thing admits somewhat of a popular explanation, if we suppose these inequalities to originate from modifications of the tangential disturbing forces: for then, if Venus should tend to draw the Earth forward in its orbit, the Earth must tend to draw Venus back: at the same time and for equal times, but not by equal degrees: since the accelerating force of Venus on the Earth, would, from her smaller mass, be less than the Earth's retarding force of Venus; but the correspondent accelerations and retardations of the mean motions will not, for obvious reasons, be necessarily proportional to the masses of the retarding and accelerating bodies. They follow, as we have seen in one case, see p. 38. a different ratio which must be ascertained by calculation. Indeed the preceding statement, as it was said in its

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