The period of Hilda (153) is more than two and a half times that of Medusa (149). This is greater than the ratio of Saturn's period to that of Jupiter. The maximum observed between 2000 and 2100 days corresponds to the space immediately interior to chasm I. on a previous page, that between 1300 and 1400 to the space interior to the second, and that between 1500 and 1700 to the part of the zone within the fourth gap. The table presents quite numerous instances of approximate equality; in forty-three cases the periods differing less than twenty-four hours. It is impossible to say, however, whether any two of these periods are _exactly_ equal. In cases of a very close approach two asteroids, notwithstanding their small mass, may exert upon each other quite sensible perturbations. 11. Origin of the Asteroids. But four minor planets had been discovered when Laplace issued his last edition of the "Système du Monde." The author, in his celebrated seventh note in the second volume of that work, explained the origin of these bodies by assuming that the primitive ring from which they were formed, instead of collecting into a single sphere, as in the case of the major planets, broke up into four distinct masses. But the form and extent of the cluster as now known, as well as the observed facts bearing on the constitution of Saturn's ring, seem to require a modification of Laplace's theory. Throughout the greater part of the interval between Mars and Jupiter an almost continuous succession of small planetary masses--not nebulous rings--appears to have been abandoned at the solar equator. The entire cluster, distributed throughout a space whose outer radius exceeds the inner by more than two hundred millions of miles, could not have originated, as supposed by Laplace, in a single nebulous zone the different parts of which revolved with the same angular velocity. The following considerations may furnish a suggestion in regard to the mode in which these bodies were separated from the equator of the solar nebula. (_a_) The perihelion distance of Jupiter is 4.950, while the aphelion distance of Hilda is 4.623. If, therefore, the sun once extended to the latter, the central attraction of its mass on an equatorial particle was but five times greater than Jupiter's perihelion influence on the same. It is easy to see, then, that this "giant planet" would produce enormous tidal elevations in the solar mass. (_b_) The centrifugal force would be greatest at the crest of this tidal wave. (_c_) Three periods of solar revolution were then about equal to two periods of Jupiter. The disturbing influence of the planet would therefore be increased at each conjunction with this protuberance. The ultimate separation (not of a ring but) of a planetary mass would be the probable result of these combined and accumulating forces. 12. Variability of Certain Asteroids.