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CHAP. VI.

Of the Perturbations of the Satellites of Jupiter.

THE first inequalities which observation discovered in the motion of these bodies, are also the first which are derived from the theory of their mutual attractions. We have seen in the second Book, that there exists

1. An equation in the motion of the first satellite equal to * 5258", multiplied by the sine of double the excess of the mean longitude of the first satellite above that of the second.

2. An equation in the motion of the second satellite equal to † 11923", multiplied by the sine of the excess of the first satellite above that of the third.

* 28' 23′′ 5. + 1° 4' 23".

3. An equation in the motion of the third satellite equal to * 827", multiplied by the sine of the excess of the longitude of the second satellite above that of the third.

Not only the theory of gravity gives these inequalities, as Lagrange and Bailly were the first to remark, but it shews us also, what observation seemed to indicate, that the inequality of the second satellite is the result of two inequalities, of which one being caused by the action of the first satellite, varies as the sine of the excess of the longitude of the first satellite above that of the second; and the other, produced by the action of the third, varies as the sine of double the excess of the longitude of the second satellite above that of the third. Thus the second satellite experiences a perturbation from the action of the first, similar to that which itself causes in the third; and it experiences from the third a similar perturbation to that which itself causes in the first.

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These two inequalities are combined into one in consequence of the relation which exists between the mean motions and the mean longitudes of the three first satellites, for the mean motion of the first satellite plus twice that of the third, is equal to three times that of the second; and the mean longitude of the first satellite minus three times that of the second plus twice that of the third is constantly equal to a semi-circumference: but will these relations always exist, or are they only approximative, and will not the two inequalities of the second satellite, at present combined, be separated in the course of time? It is to theory that we must apply for a solution to this question.

The approximation which the tables gave to the preceding relations, made me suppose that they were rigorously exact; for it was against all probability that chance should have originally placed the three first satellites at the precise distances and positions suitable to the above relation it was therefore extremely probable that it arose

from some particular cause; I looked therefore for this cause in the mutual action of the satellites. A scrupulous investigation of this action, has shewn me that it has caused these relations to be rigorously exact; from whence I concluded, that in determining again by the examination of a great many distant observations, the mean longitudes of the three first satellites, it would be found that they would approximate still more to these relations, to which the tables should be made exactly to agree. I had the satisfaction of seeing this consequence of the theory confirmed, with remarkable precision, by the researches which Delambre has lately made concerning the satellites of Jupiter. It is not necessary that these relations should have taken place exactly at their origin, it was enough that they did not greatly differ, then the mutual actions of the satellites upon each other were sufficient to subject them to this law, and to maintain it unaltered; but the little difference between this and the primitive rela tion, has given rise to a small inequality of

an arbitrary extent, and unequally distributed among the three satellites, and which Í have distinguished under the name of libration. The two constant arbitrary quantities of this inequality, replace whatever arbitrary quantity is made to disappear by the two preceding relations, in the mean motions and in the epochs of the mean longitudes of the three first satellites; for the number of arbitrary quantities included in the theory of a system of bodies is necessarily sextuple the number of bodies: as observation does not indicate this inequality, it must evidently be very small, and even insensible.

The preceding relations would still subsist, even if the mean motions of the satellites were subject. to secular variations analogous to that in the motion of the Moon. They would subsist also in the case of these motions being altered by the resistance of a medium, or by other causes, provided their effects were so small as not to be perceived in less than a century. In all these cases the secular equations so arrange themselves, by the reciprocal action

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