" 1600 " 1700 " 54 "

" 1700 " 1800 " 20 "

" 1800 " 1900 " 13 "

" 1900 " 2000 " 19 "

" 2000 " 2100 " 33 "

" 2100 " 2200 " 2 "

" 2200 " 2300 " 2 "

" 2300 " 2400 " 8 "

" 2400 " 2800 " 0 "

" 2800 " 2900 " 2 "

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 s.p.a.ce immediately interior to chasm I. on a previous page, that between 1300 and 1400 to the s.p.a.ce 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 ma.s.s, 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 "Systeme du Monde." The author, in his celebrated seventh note in the second volume of that work, explained the origin of these bodies by a.s.suming 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 ma.s.ses. But the form and extent of the cl.u.s.ter as now known, as well as the observed facts bearing on the const.i.tution 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 ma.s.ses--not nebulous rings--appears to have been abandoned at the solar equator. The entire cl.u.s.ter, distributed throughout a s.p.a.ce 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 ma.s.s 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 ma.s.s.

(_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 ma.s.s would be the probable result of these combined and acc.u.mulating forces.

12. Variability of Certain Asteroids.

Observations of some minor planets have indicated a variation of their apparent magnitudes. Frigga, discovered by Dr. Peters in 1862, was observed at the next opposition in 1864; but after this it could not be found till 1868, when it was picked up by Professor Tietjen. From the latter date its light seems again to have diminished, as all efforts to re-observe it were unsuccessful till 1879. According to Dr. Peters, the change in brightness during the period of observation in that year was greater than that due to its varying distance. No explanation of such changes has yet been offered. It has been justly remarked, however, that "the length of the period of the fluctuation does not allow of our connecting it with the rotation of the planet."

13. The Average Asteroid Orbit.

At the meeting of the American a.s.sociation for the Advancement of Science in 1884, Professor Mark W. Harrington, of Ann Arbor, Michigan, presented a paper in which the elements of the asteroid system were considered on the principle of averages. Two hundred and thirty orbits, all that had then been determined, were employed in the discussion.

Professor Harrington supposes two planes to intersect the ecliptic at right angles; one pa.s.sing through the equinoxes and the other through the solstices. These planes will intersect the asteroidal orbits, each in four points, and "the mean intersection at each solstice and equinox may be considered a point in the average orbit."

In 1883 the Royal Academy of Denmark offered its gold medal for a statistical examination of the orbits of the small planets considered as parts of a ring around the sun. The prize was awarded in 1885 to M.

Svedstrup, of Copenhagen. The results obtained by these astronomers severally are as follows:

+-----------------------------+-------------+------------+

Harrington.

Svedstrup.

+-----------------------------+-------------+------------+

Longitude of perihelion

14 39'

101 48'

" of ascending node

113 56

133 27

Inclination

1 0

6 6

Eccentricity

0.0448

0.0281

Mean distance

2.7010

2.6435

+-----------------------------+-------------+------------+

These elements, with the exception of the first, are in reasonable harmony.

14. The Relation of Short-Period Comets to the Zone of Asteroids.

Did comets originate within the solar system, or do they enter it from without? Laplace a.s.signed them an extraneous origin, and his view is adopted by many eminent astronomers. With all due respect to the authority of great names, the present writer has not wholly abandoned the theory that some comets of short period are specially related to the minor planets. According to M. Lehmann-Filhes, the eccentricity of the third comet of 1884, before its last close approach to Jupiter, was only 0.2787.[12] This is exceeded by that of twelve known minor planets. Its mean distance before this great perturbation was about 4.61, and six of its periods were nearly equal to five of Jupiter's,--a commensurability of the first order. According to Hind and Krueger, the great transformation of its...o...b..t by Jupiter's influence occurred in May, 1875. It had previously been an asteroid too remote to be seen even in perihelion. This body was discovered by M. Wolf, at Heidelberg, September 17, 1884. Its present period is about six and one-half years.

The perihelion distance of the comet 1867 II. at its return in 1885 was 2.073; its aphelion is 4.897; so that its entire path, like those of the asteroids, is included between the orbits of Mars and Jupiter. Its eccentricity, as we have seen, is little greater than that of aethra, and its period, inclination, and longitude of the ascending node are approximately the same with those of Sylvia, the eighty-seventh minor planet. In short, this comet may be regarded as an asteroid whose elements have been considerably modified by perturbation.

It has been stated that the gap at the distance 3.277 is the only one corresponding to the first order of commensurability. The distance 3.9683, where an asteroid's period would be two-thirds of Jupiter's, is immediately beyond the outer limit of the cl.u.s.ter as at present known; the mean distance of Hilda being 3.9523. The discovery of new members beyond this limit is by no means improbable. Should a minor planet at the mean distance 3.9683 attain an eccentricity of 0.3--and this is less than that of eleven now known--its aphelion would be more remote than the perihelion of Jupiter. Such an orbit might not be stable. Its form and extent might be greatly changed after the manner of Lexell's comet.

Two well-known comets, Faye's and Denning's, have periods approximately equal to two-thirds of Jupiter's. In like manner the periods of D'Arrest's and Biela's comets correspond to the hiatus at 3.51, and that of 1867 II. to that at 3.277.

Of the thirteen telescopic comets whose periods correspond to mean distances within the asteroid zone, all have direct motion; all have inclinations similar to those of the minor planets; and their eccentricities are generally less than those of other known comets. Have these facts any significance in regard to their origin?

APPENDIX.

NOTE A.

THE POSSIBLE EXISTENCE OF ASTEROIDS IN UNDISCOVERED RINGS.

If Jupiter's influence was a factor in the separation of planetules at the sun's equator, may not similar cl.u.s.ters exist in other parts of our system? The hypothesis is certainly by no means improbable. For anything we know to the contrary a group may circulate between Jupiter and Saturn; such bodies, however, could not be discovered--at least not by ordinary telescopes--on account of their distance. The Zodiacal Light, it has been suggested, may be produced by a cloud of indefinitely small particles related to the planets between the sun and Mars. The rings of Saturn are merely a dense asteroidal cl.u.s.ter; and, finally, the phenomena of luminous meteors indicate the existence of small ma.s.ses of matter moving with different velocities in interstellar s.p.a.ce.

NOTE B.

THE ORIGIN AND STRUCTURE OF COSMICAL RINGS.

The general theory of cosmical rings and of their arrangement in sections or cl.u.s.ters with intervening chasms may be briefly stated in the following propositions:

I.

Whenever the separating force of a primary body on a secondary or satellite is greater than the central attraction of the latter on its superficial stratum, the satellite, if either gaseous or liquid, will be transformed into a ring.

EXAMPLES.--Saturn's ring, and the meteoric rings of April 20, August 10, November 14, and November 27.

See Payne's _Sidereal Messenger_, April, 1885.

II.

When a cosmical body is surrounded by a ring of considerable breadth, and has also exterior satellites at such distances that a simple relation of commensurability would obtain between the periods of these satellites and those of certain particles of the ring, the disturbing influence of the former will produce gaps or intervals in the ring so disturbed.

See "Meteoric Astronomy," Chapter XII.; also the _Proceedings of the American Philosophical Society_, October 6, 1871; and the _Sidereal Messenger_ for February, 1884; where the papers referred to a.s.sign a physical cause for the gaps in Saturn's ring.

THE END.

FOOTNOTES:

[1] The discoverer, Piazzi, was not, as has been so often affirmed, one of the astronomers to whom the search had been especially committed.

[2] Ma.s.salia was discovered by De Gasparis, at Naples, Sept. 19, 1852, and independently, the next night, by Chacornac, at Ma.r.s.eilles. The name was given by the latter.