are very nearly the same as if no other disturbing force existed, so great is the effect produced by Jupiter's oblateness. (240.) The figure of Saturn, including in our consideration the ring which surrounds him, is different from that of Jupiter; but the same principles will apply to the general explanation of its effects on the motion of its satellites. The body of Saturn is oblate, and the forces which it produces are exactly similar to those produced by Jupiter. The effect of the ring may be thus conceived:-If we inscribe a spherical surface in an oblate spheroid, touching its surface at the two poles, the spheroid will be divided into two parts; a sphere whose attraction is the same as if all its matter were collected at its centre, and an equatorial protuberance analogous in form to a ring. The whole irregularity in the attraction of the spheroid is evidently due to the attraction of this ring-like protuberance, since there is no such irregularity in the attraction of the sphere. We infer, therefore, that the irregularity in the attraction of a ring is of the same kind as the irregularity in the attraction of a spheroid, but that it bears a much greater proportion to the whole attraction for the ring than for the spheroid, since the ring produces all the irregularity without the whole attraction. Now, the plane of Saturn's ring coincides with the plane of Saturn's equator, so that the effect of the body and ring together is found by simply adding effects of the same kind, and is the same as if Saturn were very oblate. The rate of progression of the perisaturnium of any satellite, and the rate of regression of its node, will, therefore, be rapid. In other respects, it is probable, that the theory of these satellites would be very simple, since all (except the sixth) appear to be very small, and the sun's disturbing force is too small to produce any sensible effects. (241.) The satellites of Saturn, except the sixth, have been observed so little, that no materials exist upon which a theory can be founded. A careful series of observations on the sixth satellite has lately been made by Bessel, from which, by comparing the observed progress of the perisaturnium and regression of the node, with those calculated on an assumed mass of the ring, the real mass of the ring has been found. It appears, thus, that the mass of the ring (supposing the whole effect due to the ring) is about th of the mass of the planet. (242.) The effect of the earth's oblateness in increasing the rapidity of regression of the moon's nodes is so small, that it cannot be discovered from observation. But the effect on the position of the fundamental plane is discoverable. We have seen (204.) that the moon's line of nodes regresses completely round in 19 years. The plane of the earth's equator is inclined 234° to the earth's orbit, and the line of intersection alters very slowly. At some time, therefore, the line of nodes coincides with the intersection of the plane of the earth's equator and the plane of the earth's orbit, so that the plane of the moon's orbit lies between those two planes; and 9 years later, the line of nodes again coincides with the same line, but the orbit is inclined the other way, so that the plane of the moon's orbit is more inclined than the plane of the earth's orbit to the plane of the earth's equator. Now it is found, that in the former case the inclination of the moon's orbit to the earth's orbit is greater than in the latter by about 16", and this shows, that the plane to which the inclination has been uniform, is neither the plane of the earth's equator, nor that of the earth's orbit, but makes with the latter an angle of about 8", and is inclined towards the former. (243.) There is another effect of the earth's oblateness (the only other effect on the moon which is sensible) that deserves notice. The inclination of the moon's orbit to the earth's orbit is less than 5°, and the inclination of the earth's equator to the earth's orbit is 231. Consequently, when the moon's orbit lies between these two planes, the inclination of the moon's orbit to the earth's equator is about 19°; and when the line of nodes is again in the same position, but the orbit is inclined the other way, the inclination of the moon's orbit to the earth's equator is about 28°. At the latter time, therefore, in consequence of the earth's oblateness, the moon, when farthest from its node, will, by (235.), experience a smaller attraction to the earth than at the former time when farthest from its node. When in the line of nodes, the attractions in the two cases will be equal. On the whole, therefore, the attraction to the earth will be less at the latter time than at the former. For the period of 93 years, therefore, the earth's attraction on the moon is gradually diminished, and then is gradually increased for the same time. The moon's orbit (47.) becomes gradually larger in the first of these times, and smaller in the second. The change is very minute, but, as explained in (49.), the alteration in the longitude may be sensible. It is found by observation to amount to about 8", by which the moon is sometimes before her mean place, and sometimes behind it. If the earth's flattening at each pole were more or less thanth of the semi-diameter, the effects on the moon, both in altering the position of the fundamental plane, and in producing this inequality in the longitude, would be greater or less than the quantities that we have mentioned; and thus we are led to the very remarkable conclusion, that by observing the moon we can discover the amount of the earth's oblateness, supposing the theory to be true. This has been done; and the agreement of the result thus obtained, with that obtained from direct measures of the earth, is one of the most striking proofs of the correctness of the Theory of Universal Gravitation. THE END. |