imagined, and which must necessarily belong to a refined state of analytical science.

In a subsequent Chapter of this Work, the variations of the elements of a planet's orbit, or its secular inequalities will be treated of by a more direct method than the one that has been just described. We will now consider whether there are any periodical inequalities other than those already investigated, that claim our attention. We shall find such in the theory of Jupiter and Saturn.

The perturbations of these planets, require, like those of Venus and the Earth, that special or peculiar computation which has been described in pp. 286, &c.

It would seem then that no cases could be more alike than the preceding; and that the solution of the Problem of the three Bodies, for Venus the Earth and Sun, would be, virtually and in substance, the just solution, when Jupiter, Saturn and the Sun should be the three bodies.

But here, as frequently in intricate investigations, it happens that general views and analogies are altogether fallacious. The theory of the perturbations of Jupiter and Saturn contains very distinct peculiarities. It differs, in certain respects, not only from that of the perturbations of Venus and the Earth, but from every other planetary theory. The only points of resemblance to it are to be found in the system of Jupiter's Satellites. But we will proceed to explain, without farther preamble, in what the peculiarities above alluded to consist.

CHAP. XIX.

On certain Inequalities of Jupiter and Saturn, which depend on the near Commensurability of their Mean Motions. Five times Saturn's Mean Motion nearly equal to twice Jupiter's. The peculiar Inequalities of Jupiter and Saturn expounded by Terms involving the Cubes of the Eccentricities. The Cause of their magnitude. Connexion, in the same Term, between the Power of the Eccentricity and the Form of the Argument. Expressions for the Retardation of Saturn, and the corresponding Acceleration of Jupiter. Agreement of the Results of Computation and Observation. Period of the Inequality. A similar Inequality in the Motion of Mercury, &c. &c.

THE condition that renders singular the case of the mutual perturbations of Jupiter and Saturn, is the numerical relation that subsists between their mean distances; which is such that the mean motions of the two planets are to one another almost in a definite proportion.

If we examine the value of R, and the forms of the integrals by which the longitude and parallax are expressed, we shall easily perceive what kind of peculiarity of result must ensue, if the mean motion of the disturbed should be to n' the mean motion of the disturbing planet nearly as number is to number.

Take the most simple case: suppose n to be to n' nearly as 1 to 2 now, one of the terms of R (see p. 279.) is of the form

and, in consequence of this term and corresponding to it, there will be introduced into d R a term such as

and, accordingly, into the value of dv, and by virtue of the term 3affn dt.d R which it contains, this term

Now 2nn' is by supposition very small: the coefficient, therefore, of the above term would receive from the divisor (2n n') (by as much indeed as that divisor can confer) considerable magnitude, and the term, in its resulting value, notwithstanding the minuteness of P and e, might expound an equation of considerable moment.

The magnitude of the equation is not the sole consequence of the minuteness of 2 n-n. The period of the equation, as it is plain from pp. 235, 236: would be increased by it, and become greater the smaller 2 n n' should be.

The case we have put is altogether hypothetical: amongst the planets there are no two whose mean motions are either as 2 to 1, or nearly so*.

But if n' should be nearly to n either as 3 to 2, or, as 3 to 1, or, as 5 to 2, or, as 4 to 1, or as, &c., or generally as i' to i, there would arise, in any one of these cases, an equation of some magnitude and with a long period: the length of the latter depending on the minuteness of i'n'-in: the magnitude of the former depending partly on that condition and on other conditions.

The first point is easily made out; if we revert to the note of p. 235, it will appear that the period of the equation, or that interval of time in which it will pass through all its degrees and affections of magnitude, will be the larger the smaller i'nin is: but the magnitude of the coefficient (which is the greatest value of the equation) must depend partly on that of e, or, that being given, on the power of e which it involves.

This brings us to the very jet of the business: the term in the differential equation may involve e, or e, or, e, and, in con

* The mean motions of the first and second, of the second and third Satellites of Jupiter, are, however, in that proportion.

sequence thereof, may be extremely small: but the corresponding term in the integrated equation, may, by having received a small divisor, become of some value: in other words, a very small modification of the disturbing force, may, by the duration of its agency, or by the accumulation of its effects, sensibly affect the disturbed planet's place.

The terms likely to be neglected by the computist would be those involving the squares and cubes of the eccentricity. 'Nous pouvons (says Euler in an ineffectual Essay to explain the irregularities of Jupiter and Saturn) hardiment negliger les termes qui renferment le quarré et les plus hautes puissances de l'eccentricité.' The cube of the eccentricity of Jupiter's orbit is .00011183: the terms, therefore, that involve both this fraction and the fraction expounding the disturbing force must be extremely small in the differential equation. They are the very terms, however, as we shall soon see, that require, in the theory of Jupiter and Saturn, particular consideration.

The very minute modifications of the disturbing force, expounded by such terms, can produce effects only in one way; that is, by the great duration of their agency: in other words, their periods must be very large: if, therefore,

Pe. cos. (in't intié - ie),

should represent one of the above-mentioned terms, it would follow (see p. 235.) that

i'n' - in,

must be a very small quantity. :

The terms then in the differential equation that are extremely small from involving e3, and the quantity expounding the disturbing force, may become of moment from receiving by integration (which is the scientific summation of small terms) divisors such as in in, or (i'nin2. But there is no necessary connexion whatever between the minuteness of i'n'-in, and that of e3.

If we wish to know, antecedently to actual computation, whether the mean motions n' and n are so related, that, i' and i being two integers, i'n'-in can be either nothing or nearly so:

we must examine the numbers expressing the mean motions and make trial with them. Now in the case of Jupiter and Saturn, n' (h's mean annual motion) = 43996′′.72

n (4's mean annual motion) = 109256".23

if therefore we take i' 5, and i

= 2, we shall have 5n2n 1471".14,

a small quantity relatively to n or n': and these integers 5 and 2, are the only ones, having a difference equal to 3, that make i'n' in a small quantity. In the other planets, whatever be the two selected, there are no two integers i' and i (having a difference either 1, 2, 3 or 4) that make i'n' in a quantity equally small with the preceding.

But we have not yet shewn what the terms are that have the argument 5 n' 2n the fact is, such terms involve the cube of the eccentricity, and on that account are extremely small: but they expound a modification of the disturbing force, the agency of which, either continually accelerating, or continually retarding the body's motion, must endure for a very long time: for, since 5n2n = 1471", the whole period of its action (see p. 235.) is about 900 years.

Having thus ascertained, by antecedent considerations, the existence of a very small inequality of a very long period, let us consider in what manner it would affect the phenomena of observation and their determination.

The mean motion of a planet (see Astron. p. 263.) is determined by observing the planet in two similar oppositions, and then by dividing the interval of time by the number of revolutions.

Now, an inequality, such as has been described, acting almost by insensible degrees, and for a great length of time, would affect the determination of the mean motion. Its effects would be blended with it; and without the aid of theory it would be impossible to disengage them. For, if the annual effect of the inequality should not exceed a few seconds, and its period should be 900 years, no comparison of observations, made during an