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Transcriber's note:

      Text enclosed by underscores is in italics (_italics_).

      [pi] represents the Greek letter pi, and [AN] the symbol for
      the ascending node. The degree sign has been replaced by D
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      asteroids in Tables I and II have been corrected.





THE ASTEROIDS,

Or Minor Planets Between Mars and Jupiter.

by

DANIEL KIRKWOOD, LL.D.,

Professor Emeritus in the University of Indiana; Author of "Comets
and Meteors," "Meteoric Astronomy," etc.







Philadelphia:
J. B. Lippincott Company.
1888.

Copyright, 1887, by Daniel Kirkwood.

[Illustration]




PREFACE.


The rapid progress of discovery in the zone of minor planets, the
anomalous forms and positions of their orbits, the small size as well as
the great number of these telescopic bodies, and their peculiar
relations to Jupiter, the massive planet next exterior,--all entitle
this part of the system to more particular consideration than it has
hitherto received. The following essay is designed, therefore, to supply
an obvious want. Its results are given in some detail up to the date of
publication. Part I. presents in a popular form the leading historical
facts as to the discovery of Ceres, Pallas, Juno, Vesta, and Astraea; a
tabular statement of the dates and places of discovery for the entire
group; a list of the names of discoverers, with the number of minor
planets detected by each; and a table of the principal elements so far
as computed.

In Part II. this descriptive summary is followed by questions relating
to the origin of the cluster; the elimination of members from particular
parts; the eccentricities and inclinations of the orbits; and the
relation of the zone to comets of short period. The elements are those
given in the Paris _Annuaire_ for 1887, or in recent numbers of the
_Circular zum Berliner Astronomischen Jahrbuch_.

DANIEL KIRKWOOD.

BLOOMINGTON, INDIANA, November, 1887.




CONTENTS.


 PART I.                                                       PAGE

 PLANETARY DISCOVERIES BEFORE THE ASTEROIDS WERE KNOWN            9

 DISCOVERY OF THE FIRST ASTEROIDS                                11

 TABLE I.--ASTEROIDS IN THE ORDER OF THEIR DISCOVERY             17

 NUMBERS FOUND BY THE RESPECTIVE DISCOVERERS                     23

 NUMBERS DISCOVERED IN THE DIFFERENT MONTHS                      25

 MODE OF DISCOVERY                                               25

 NAMES AND SYMBOLS                                               25

 MAGNITUDES OF THE ASTEROIDS                                     26

 ORBITS OF THE ASTEROIDS                                         28

 TABLE II.--ELEMENTS OF THE ASTEROIDS                            29


 PART II.

 EXTENT OF THE ZONE                                              37

 THEORY OF OLBERS                                                38

 SMALL MASS OF THE ASTEROIDS                                     38

 LIMITS OF PERIHELION DISTANCE                                   39

 DISTRIBUTION OF THE ASTEROIDS IN SPACE                          40

 LAW OF GAP FORMATION                                            42

 COMMENSURABILITY OF PERIODS WITH THAT OF JUPITER                43

 ORDERS OF COMMENSURABILITY                                      44

 ELIMINATION OF VERY ECCENTRIC ORBITS                            46

 RELATIONS BETWEEN CERTAIN ADJACENT ORBITS                       47

 THE ECCENTRICITIES                                              48

 THE INCLINATIONS                                                49

 LONGITUDES OF THE PERIHELIA AND OF THE ASCENDING NODES          50

 THE PERIODS                                                     51

 ORIGIN OF THE ASTEROIDS                                         52

 VARIABILITY OF CERTAIN ASTEROIDS                                53

 THE AVERAGE ASTEROID ORBIT                                      54

 THE RELATION OF SHORT-PERIOD COMETS TO THE ZONE OF ASTEROIDS    55

 APPENDIX                                                        59




PART I.




THE ASTEROIDS, OR MINOR PLANETS BETWEEN MARS AND JUPITER.




1. Introductory.


PLANETARY DISCOVERIES BEFORE THE ASTEROIDS WERE KNOWN.

The first observer who watched the skies with any degree of care could
not fail to notice that while the greater number of stars maintained the
same relative places, a few from night to night were ever changing their
positions. The planetary character of Mercury, Venus, Mars, Jupiter, and
Saturn was thus known before the dawn of history. The names, however, of
those who first distinguished them as "wanderers" are hopelessly lost.
Venus, the morning and evening star, was long regarded as two distinct
bodies. The discovery that the change of aspect was due to a single
planet's change of position is ascribed to Pythagoras.

At the beginning of the seventeenth century but six primary planets and
one satellite were known as members of the solar system. Very few, even
of the learned, had then accepted the theory of Copernicus; in fact,
before the invention of the telescope the evidence in its favor was not
absolutely conclusive. On the 7th of January, 1610, Galileo first saw
the satellites of Jupiter. The bearing of this discovery on the theory
of the universe was sufficiently obvious. Such was the prejudice,
however, against the Copernican system that some of its opponents denied
even the reality of Galileo's discovery. "Those satellites," said a
Tuscan astronomer, "are invisible to the naked eye, and therefore can
exercise no influence on the earth, and therefore would be useless, and
therefore do not exist. Besides, the Jews and other ancient nations, as
well as modern Europeans, have adopted the division of the week into
_seven_ days, and have named them from the seven planets; now, if we
increase the number of planets this whole system falls to the ground."

No other secondary planet was discovered till March 25, 1655, when
Titan, the largest satellite of Saturn, was detected by Huyghens. About
two years later (December 7, 1657) the same astronomer discovered the
true form of Saturn's ring; and before the close of the century
(1671-1684) four more satellites, Japetus, Rhea, Tethys, and Dione, were
added to the Saturnian system by the elder Cassini. Our planetary
system, therefore, as known at the close of the seventeenth century,
consisted of six primary and ten secondary planets.

Nearly a century had elapsed from the date of Cassini's discovery of
Dione, when, on the 13th of March, 1781, Sir William Herschel
enlarged the dimensions of our system by the detection of a
planet--Uranus--exterior to Saturn. A few years later (1787-1794) the
same distinguished observer discovered the first and second satellites
of Saturn, and also the four Uranian satellites. He was the only planet
discoverer of the eighteenth century.


2. Discovery of the First Asteroids.

As long ago as the commencement of the seventeenth century the
celebrated Kepler observed that the respective distances of the planets
from the sun formed nearly a regular progression. The series, however,
by which those distances were expressed required the interpolation of a
term between Mars and Jupiter,--a fact which led the illustrious German
to predict the discovery of a planet in that interval. This conjecture
attracted but little attention till after the discovery of Uranus, whose
distance was found to harmonize in a remarkable manner with Kepler's
order of progression. Such a coincidence was of course regarded with
considerable interest. Towards the close of the last century Professor
Bode, who had given the subject much attention, published the law of
distances which bears his name, but which, as he acknowledged, is due to
Professor Titius. According to this formula the distances of the planets
from Mercury's orbit form a geometrical series of which the ratio is
two. In other words, if we reckon the distances of Venus, the earth,
etc., from the orbit of Mercury, instead of from the sun, we find
that--interpolating a term between Mars and Jupiter--the distance of any
member of the system is very nearly half that of the next exterior.
Baron De Zach, an enthusiastic astronomer, was greatly interested in
Bode's empirical scheme, and undertook to determine the elements of the
hypothetical planet. In 1800 a number of astronomers met at Lilienthal,
organized an astronomical society, and assigned one twenty-fourth part
of the zodiac to each of twenty-four observers, in order to detect, if
possible, the unseen planet. When it is remembered that at this time no
primary planet had been discovered within the ancient limits of the
solar system, that the object to be looked for was comparatively near
us, and that the so-called law of distances was purely empirical, the
prospect of success, it is evident, was extremely uncertain. How long
the watch, if unsuccessful, might have been continued is doubtful. The
object of research, however, was fortunately brought to light before the
members of the astronomical association had fairly commenced their
labors.[1]

On the 1st of January, 1801, Professor Giuseppe Piazzi, of Palermo,
noticed a star of the eighth magnitude, not indicated in Wollaston's
catalogue. Subsequent observations soon revealed its planetary
character, its mean distance corresponding very nearly with the
calculations of De Zach. The discoverer called it Ceres Ferdinandea, in
honor of his sovereign, the King of Naples. In this, however, he was not
followed by astronomers, and the planet is now known by the name of
Ceres alone. The discovery of this body was hailed by astronomers with
the liveliest gratification as completing the harmony of the system.
What, then, was their surprise when in the course of a few months this
remarkable order was again interrupted! On the 28th of March, 1802, Dr.
William Olbers, of Bremen, while examining the relative positions of the
small stars along the path of Ceres, in order to find that planet with
the greater facility, noticed a star of the seventh or eighth magnitude,
forming with two others an equilateral triangle where he was certain no
such configuration existed a few months before. In the course of a few
hours its motion was perceptible, and on the following night it had very
sensibly changed its position with respect to the neighboring stars.
Another planet was therefore detected, and Dr. Olbers immediately
communicated his discovery to Professor Bode and Baron De Zach. In his
letter to the former he suggested Pallas as the name of the new member
of the system,--a name which was at once adopted. Its orbit, which was
soon computed by Gauss, was found to present several striking anomalies.
The inclination of its plane to that of the ecliptic was nearly
thirty-five degrees,--an amount of deviation altogether extraordinary.
The eccentricity also was greater than in the case of any of the old
planets. These peculiarities, together with the fact that the mean
distances of Ceres and Pallas were nearly the same, and that their
orbits approached very near each other at the intersection of their
planes, suggested the hypothesis that they are fragments of a single
original planet, which, at a very remote epoch, was disrupted by some
mysterious convulsion. This theory will be considered when we come to
discuss the tabulated elements of the minor planets now known.

For the convenience of astronomers, Professor Harding, of Lilienthal,
undertook the construction of charts of all the small stars near the
orbits of Ceres and Pallas. On the evening of September 1, 1804, while
engaged in observations for this purpose, he noticed a star of the
eighth magnitude not mentioned in the great catalogue of Lalande. This
proved to be a third member of the group of asteroids. The discovery was
first announced to Dr. Olbers, who observed the planet at Bremen on the
evening of September 7.

Before Ceres had been generally adopted by astronomers as the name of
the first asteroid, Laplace had expressed a preference for Juno. This,
however, the discoverer was unwilling to accept. Mr. Harding, like
Laplace, deeming it appropriate to place Juno near Jupiter, selected the
name for the third minor planet, which is accordingly known by this
designation.

Juno is distinguished among the first asteroids by the great
eccentricity of its orbit, amounting to more than 0.25. Its least and
its greatest distances from the sun are therefore to each other very
nearly in the ratio of three to five. The planet consequently receives
nearly three times as much light and heat in perihelion as in aphelion.
It follows, also, that the half of the orbit nearest the sun is
described in about eighteen months, while the remainder, or more distant
half, is not passed over in much less than three years. Schroeter
noticed a variation in the light of Juno, which he supposed to be
produced by an axial rotation in about twenty-seven hours.

The fact that Juno was discovered not far from the point at which the
orbit of Pallas approaches very near that of Ceres, was considered a
strong confirmation of the hypothesis that the asteroids were produced
by the explosion of a large planet; for in case this hypothesis be
founded in truth, it is evident that whatever may have been the forms of
the various orbits assumed by the fragments, they must all return to the
point of separation. In order, therefore, to detect other members of the
group, Dr. Olbers undertook a systematic examination of the two opposite
regions of the heavens through which they must pass. This search was
prosecuted with great industry and perseverance till ultimately crowned
with success. On the 29th of March, 1807, while sweeping over one of
those regions through which the orbits of the known asteroids passed, a
star of the sixth magnitude was observed where none had been seen at
previous examinations. Its planetary character, which was immediately
suspected, was confirmed by observation, its motion being detected on
the very evening of its discovery. This fortunate result afforded the
first instance of the discovery of two primary planets by the same
observer.

The astronomer Gauss having been requested to name the new planet, fixed
upon Vesta, a name universally accepted. Though the brightest of the
asteroids, its apparent diameter is too small to be accurately
determined, and hence its real magnitude is not well ascertained.
Professor Harrington, of Ann Arbor, has estimated the diameter at five
hundred and twenty miles. According to others, however, it does not
exceed three hundred. If the latter be correct, the volume is about
1/20000 that of the earth. It is remarkable that notwithstanding its
diminutive size it may be seen under favorable circumstances by the
naked eye.

Encouraged by the discovery of Vesta (which he regarded as almost a
demonstration of his favorite theory), Dr. Olbers continued his
systematic search for other planetary fragments. Not meeting, however,
with further success, he relinquished his observations in 1816. His
failure, it may here be remarked, was doubtless owing to the fact that
his examination was limited to stars of the seventh and eighth
magnitudes.

The search for new planets was next resumed about 1831, by Herr Hencke,
of Driessen. With a zeal and perseverance worthy of all praise, this
amateur astronomer employed himself in a strict examination of the
heavens represented by the Maps of the Berlin Academy. These maps extend
fifteen degrees on each side of the equator, and contain all stars down
to the ninth magnitude and many of the tenth. Dr. Hencke rendered some
of these charts still more complete by the insertion of smaller stars;
or rather, "made for himself special charts of particular districts." On
the evening of December 8, 1845, he observed a star of the ninth
magnitude where none had been previously seen, as he knew from the fact
that it was neither found on his own chart nor given on that of the
Academy. On the next morning he wrote to Professors Encke and Schumacher
informing them of his supposed discovery. "It is very improbable," he
remarked in his letter to the latter, "that this should prove to be
merely a variable star, since in my former observations of this region,
which have been continued for many years, I have never detected the
slightest trace of it." The new star was soon seen at the principal
observatories of Europe, and its planetary character satisfactorily
established. The selection of a name was left by the discoverer to
Professor Encke, who chose that of Astraea.

The facts in regard to the very numerous subsequent discoveries may best
be presented in a tabular form.


TABLE I.

_The Asteroids in the Order of their Discovery._

 -----------------+----------------+---------------+------------
   Asteroids.     | Date of        | Name of       | Place of
                  | Discovery.     | Discoverer.   | Discovery.
 -----------------+----------------+---------------+------------
   1. Ceres       | 1801, Jan. 1   | Piazzi        | Palermo
   2. Pallas      | 1802, Mar. 28  | Olbers        | Bremen
   3. Juno        | 1804, Sept. 1  | Harding       | Lilienthal
   4. Vesta       | 1807, Mar. 29  | Olbers        | Bremen
   5. Astraea     | 1845, Dec. 8   | Hencke        | Driessen
   6. Hebe        | 1847, July 1   | Hencke        | Driessen
   7. Iris        | 1847, Aug. 14  | Hind          | London
   8. Flora       | 1847, Oct. 18  | Hind          | London
   9. Metis       | 1848, Apr. 26  | Graham        | Markree
  10. Hygeia      | 1849, Apr. 12  | De Gasparis   | Naples
  11. Parthenope  | 1850, May 11   | De Gasparis   | Naples
  12. Victoria    | 1850, Sept. 13 | Hind          | London
  13. Egeria      | 1850, Nov. 2   | De Gasparis   | Naples
  14. Irene       | 1851, May 19   | Hind          | London
  15. Eunomia     | 1851, July 29  | De Gasparis   | Naples
  16. Psyche      | 1852, Mar. 17  | De Gasparis   | Naples
  17. Thetis      | 1852, Apr. 17  | Luther        | Bilk
  18. Melpomene   | 1852, June 24  | Hind          | London
  19. Fortuna     | 1852, Aug. 22  | Hind          | London
  20. Massalia    | 1852, Sept. 19 | De Gasparis   | Naples
  21. Lutetia     | 1852, Nov. 15  | Goldschmidt   | Paris
  22. Calliope    | 1852, Nov. 16  | Hind          | London
  23. Thalia      | 1852, Dec. 15  | Hind          | London
  24. Themis      | 1853, Apr. 5   | De Gasparis   | Naples
  25. Phocea      | 1853, Apr. 6   | Chacornac     | Marseilles
  26. Proserpine  | 1853, May 5    | Luther        | Bilk
  27. Euterpe     | 1853, Nov. 8   | Hind          | London
  28. Bellona     | 1854, Mar. 1   | Luther        | Bilk
  29. Amphitrite  | 1854, Mar. 1   | Marth         | London
  30. Urania      | 1854, July 22  | Hind          | London
  31. Euphrosyne  | 1854, Sept. 1  | Ferguson      | Washington
  32. Pomona      | 1854, Oct. 26  | Goldschmidt   | Paris
  33. Polyhymnia  | 1854, Oct. 28  | Chacornac     | Paris
  34. Circe       | 1855, Apr. 6   | Chacornac     | Paris
  35. Leucothea   | 1855, Apr. 19  | Luther        | Bilk
  36. Atalanta    | 1855, Oct. 5   | Goldschmidt   | Paris
  37. Fides       | 1855, Oct. 5   | Luther        | Bilk
  38. Leda        | 1856, Jan. 12  | Chacornac     | Paris
  39. Laetitia    | 1856, Feb. 8   | Chacornac     | Paris
  40. Harmonia    | 1856, Mar. 31  | Goldschmidt   | Paris
  41. Daphne      | 1856, May 22   | Goldschmidt   | Paris
  42. Isis        | 1856, May 23   | Pogson        | Oxford
  43. Ariadne     | 1857, Apr. 15  | Pogson        | Oxford
  44. Nysa        | 1857, May 27   | Goldschmidt   | Paris
  45. Eugenia     | 1857, June 27  | Goldschmidt   | Paris
  46. Hestia      | 1857, Aug. 16  | Pogson        | Oxford
  47. Aglaia      | 1857, Sept. 15 | Luther        | Bilk
  48. Doris       | 1857, Sept. 19 | Goldschmidt   | Paris
  49. Pales       | 1857, Sept. 19 | Goldschmidt   | Paris
  50. Virginia    | 1857, Oct. 4   | Ferguson      | Washington
  51. Nemausa     | 1858, Jan. 22  | Laurent       | Nismes
  52. Europa      | 1858, Feb. 4   | Goldschmidt   | Paris
  53. Calypso     | 1858, Apr. 4   | Luther        | Bilk
  54. Alexandra   | 1858, Sept. 10 | Goldschmidt   | Paris
  55. Pandora     | 1858, Sept. 10 | Searle        | Albany
  56. Melete      | 1857, Sept. 9  | Goldschmidt   | Paris
  57. Mnemosyne   | 1859, Sept. 22 | Luther        | Bilk
  58. Concordia   | 1860, Mar. 24  | Luther        | Bilk
  59. Olympia     | 1860, Sept. 12 | Chacornac     | Paris
  60. Echo        | 1860, Sept. 16 | Ferguson      | Washington
  61. Danae       | 1860, Sept. 9  | Goldschmidt   | Paris
  62. Erato       | 1860, Sept. 14 | Foerster and  | Berlin
                  |                |   Lesser      |
  63. Ausonia     | 1861, Feb. 10  | De Gasparis   | Naples
  64. Angelina    | 1861, Mar. 4   | Tempel        | Marseilles
  65. Maximiliana | 1861, Mar. 8   | Tempel        | Marseilles
  66. Maia        | 1861, Apr. 9   | Tuttle        | Cambridge, U.S.
  67. Asia        | 1861, Apr. 17  | Pogson        | Madras
  68. Leto        | 1861, Apr. 29  | Luther        | Bilk
  69. Hesperia    | 1861, Apr. 29  | Schiaparelli  | Milan
  70. Panopea     | 1861, May 5    | Goldschmidt   | Paris
  71. Niobe       | 1861, Aug. 13  | Luther        | Bilk
  72. Feronia     | 1862, May 29   | Peters and    | Clinton
                  |                |   Safford     |
  73. Clytie      | 1862, Apr. 7   | Tuttle        | Cambridge
  74. Galatea     | 1862, Aug. 29  | Tempel        | Marseilles
  75. Eurydice    | 1862, Sept. 22 | Peters        | Clinton
  76. Freia       | 1862, Oct. 21  | D'Arrest      | Copenhagen
  77. Frigga      | 1862, Nov. 12  | Peters        | Clinton
  78. Diana       | 1863, Mar. 15  | Luther        | Bilk
  79. Eurynome    | 1863, Sept. 14 | Watson        | Ann Arbor
  80. Sappho      | 1864, May 2    | Pogson        | Madras
  81. Terpsichore | 1864, Sept. 30 | Tempel        | Marseilles
  82. Alcmene     | 1864, Nov. 27  | Luther        | Bilk
  83. Beatrix     | 1865, Apr. 26  | De Gasparis   | Naples
  84. Clio        | 1865, Aug. 25  | Luther        | Bilk
  85. Io          | 1865, Sept. 19 | Peters        | Clinton
  86. Semele      | 1866, Jan. 14  | Tietjen       | Berlin
  87. Sylvia      | 1866, May 16   | Pogson        | Madras
  88. Thisbe      | 1866, June 15  | Peters        | Clinton
  89. Julia       | 1866, Aug. 6   | Stephan       | Marseilles
  90. Antiope     | 1866, Oct. 1   | Luther        | Bilk
  91. Aegina      | 1866, Nov. 4   | Borelly       | Marseilles
  92. Undina      | 1867, July 7   | Peters        | Clinton
  93. Minerva     | 1867, Aug. 24  | Watson        | Ann Arbor
  94. Aurora      | 1867, Sept. 6  | Watson        | Ann Arbor
  95. Arethusa    | 1867, Nov. 24  | Luther        | Bilk
  96. Aegle       | 1868, Feb. 17  | Coggia        | Marseilles
  97. Clotho      | 1868, Feb. 17  | Coggia        | Marseilles
  98. Ianthe      | 1868, Apr. 18  | Peters        | Clinton
  99. Dike        | 1868, May 28   | Borelly       | Marseilles
 100. Hecate      | 1868, July 11  | Watson        | Ann Arbor
 101. Helena      | 1868, Aug. 15  | Watson        | Ann Arbor
 102. Miriam      | 1868, Aug. 22  | Peters        | Clinton
 103. Hera        | 1868, Sept. 7  | Watson        | Ann Arbor
 104. Clymene     | 1868, Sept. 13 | Watson        | Ann Arbor
 105. Artemis     | 1868, Sept. 16 | Watson        | Ann Arbor
 106. Dione       | 1868, Oct. 10  | Watson        | Ann Arbor
 107. Camilla     | 1868, Nov. 17  | Pogson        | Madras
 108. Hecuba      | 1869, Apr. 2   | Luther        | Bilk
 109. Felicitas   | 1869, Oct. 9   | Peters        | Clinton
 110. Lydia       | 1870, Apr. 19  | Borelly       | Marseilles
 111. Ate         | 1870, Aug. 14  | Peters        | Clinton
 112. Iphigenia   | 1870, Sept. 19 | Peters        | Clinton
 113. Amalthea    | 1871, Mar. 12  | Luther        | Bilk
 114. Cassandra   | 1871, July 23  | Peters        | Clinton
 115. Thyra       | 1871, Aug. 6   | Watson        | Ann Arbor
 116. Sirona      | 1871, Sept. 8  | Peters        | Clinton
 117. Lomia       | 1871, Sept. 12 | Borelly       | Marseilles
 118. Peitho      | 1872, Mar. 15  | Luther        | Bilk
 119. Althea      | 1872, Apr. 3   | Watson        | Ann Arbor
 120. Lachesis    | 1872, Apr. 10  | Borelly       | Marseilles
 121. Hermione    | 1872, May 12   | Watson        | Ann Arbor
 122. Gerda       | 1872, July 31  | Peters        | Clinton
 123. Brunhilda   | 1872, July 31  | Peters        | Clinton
 124. Alceste     | 1872, Aug. 23  | Peters        | Clinton
 125. Liberatrix  | 1872, Sept. 11 | Prosper Henry | Paris
 126. Velleda     | 1872, Nov. 5   | Paul Henry    | Paris
 127. Johanna     | 1872, Nov. 5   | Prosper Henry | Paris
 128. Nemesis     | 1872, Nov. 25  | Watson        | Ann Arbor
 129. Antigone    | 1873, Feb. 5   | Peters        | Clinton
 130. Electra     | 1873, Feb. 17  | Peters        | Clinton
 131. Vala        | 1873, May 24   | Peters        | Clinton
 132. Aethra      | 1873, June 13  | Watson        | Ann Arbor
 133. Cyrene      | 1873, Aug. 16  | Watson        | Ann Arbor
 134. Sophrosyne  | 1873, Sept. 27 | Luther        | Bilk
 135. Hertha      | 1874, Feb. 18  | Peters        | Clinton
 136. Austria     | 1874, Mar. 18  | Palisa        | Pola
 137. Meliboea    | 1874, Apr. 21  | Palisa        | Pola
 138. Tolosa      | 1874, May 19   | Perrotin      | Toulouse
 139. Juewa       | 1874, Oct. 10  | Watson        | Pekin
 140. Siwa        | 1874, Oct. 13  | Palisa        | Pola
 141. Lumen       | 1875, Jan. 13  | Paul Henry    | Paris
 142. Polana      | 1875, Jan. 28  | Palisa        | Pola
 143. Adria       | 1875, Feb. 23  | Palisa        | Pola
 144. Vibilia     | 1875, June 3   | Peters        | Clinton
 145. Adeona      | 1875, June 3   | Peters        | Clinton
 146. Lucina      | 1875, June 8   | Borelly       | Marseilles
 147. Protogenea  | 1875, July 10  | Schulhof      | Vienna
 148. Gallia      | 1875, Aug. 7   | Prosper Henry | Paris
 149. Medusa      | 1875, Sept. 21 | Perrotin      | Toulouse
 150. Nuwa        | 1875, Oct. 18  | Watson        | Ann Arbor
 151. Abundantia  | 1875, Nov. 1   | Palisa        | Pola
 152. Atala       | 1875, Nov. 2   | Paul Henry    | Paris
 153. Hilda       | 1875, Nov. 2   | Palisa        | Pola
 154. Bertha      | 1875, Nov. 4   | Prosper Henry | Paris
 155. Scylla      | 1875, Nov. 8   | Palisa        | Pola
 156. Xantippe    | 1875, Nov. 22  | Palisa        | Pola
 157. Dejanira    | 1875, Dec. 1   | Borelly       | Marseilles
 158. Coronis     | 1876, Jan. 4   | Knorre        | Berlin
 159. Aemilia     | 1876, Jan. 26  | Paul Henry    | Paris
 160. Una         | 1876, Feb. 20  | Peters        | Clinton
 161. Athor       | 1876, Apr. 19  | Watson        | Ann Arbor
 162. Laurentia   | 1876, Apr. 21  | Prosper Henry | Paris
 163. Erigone     | 1876, Apr. 26  | Perrotin      | Toulouse
 164. Eva         | 1876, July 12  | Paul Henry    | Paris
 165. Loreley     | 1876, Aug. 9   | Peters        | Clinton
 166. Rhodope     | 1876, Aug. 15  | Peters        | Clinton
 167. Urda        | 1876, Aug. 28  | Peters        | Clinton
 168. Sibylla     | 1876, Sept. 27 | Watson        | Ann Arbor
 169. Zelia       | 1876, Sept. 28 | Prosper Henry | Paris
 170. Maria       | 1877, Jan. 10  | Perrotin      | Toulouse
 171. Ophelia     | 1877, Jan. 13  | Borelly       | Marseilles
 172. Baucis      | 1877, Feb. 5   | Borelly       | Marseilles
 173. Ino         | 1877, Aug. 1   | Borelly       | Marseilles
 174. Phaedra     | 1877, Sept. 2  | Watson        | Ann Arbor
 175. Andromache  | 1877, Oct. 1   | Watson        | Ann Arbor
 176. Idunna      | 1877, Oct. 14  | Peters        | Clinton
 177. Irma        | 1877, Nov. 5   | Paul Henry    | Paris
 178. Belisana    | 1877, Nov. 6   | Palisa        | Pola
 179. Clytemnestra| 1877, Nov. 11  | Watson        | Ann Arbor
 180. Garumna     | 1878, Jan. 29  | Perrotin      | Toulouse
 181. Eucharis    | 1878, Feb. 2   | Cottenot      | Marseilles
 182. Elsa        | 1878, Feb. 7   | Palisa        | Pola
 183. Istria      | 1878, Feb. 8   | Palisa        | Pola
 184. Deiopea     | 1878, Feb. 28  | Palisa        | Pola
 185. Eunice      | 1878, Mar. 1   | Peters        | Clinton
 186. Celuta      | 1878, Apr. 6   | Prosper Henry | Paris
 187. Lamberta    | 1878, Apr. 11  | Coggia        | Marseilles
 188. Menippe     | 1878, June 18  | Peters        | Clinton
 189. Phthia      | 1878, Sept. 9  | Peters        | Clinton
 190. Ismene      | 1878, Sept. 22 | Peters        | Clinton
 191. Kolga       | 1878, Sept. 30 | Peters        | Clinton
 192. Nausicaa    | 1879, Feb. 17  | Palisa        | Pola
 193. Ambrosia    | 1879, Feb. 28  | Coggia        | Marseilles
 194. Procne      | 1879, Mar. 21  | Peters        | Clinton
 195. Euryclea    | 1879, Apr. 22  | Palisa        | Pola
 196. Philomela   | 1879, May 14   | Peters        | Clinton
 197. Arete       | 1879, May 21   | Palisa        | Pola
 198. Ampella     | 1879, June 13  | Borelly       | Marseilles
 199. Byblis      | 1879, July 9   | Peters        | Clinton
 200. Dynamene    | 1879, July 27  | Peters        | Clinton
 201. Penelope    | 1879, Aug. 7   | Palisa        | Pola
 202. Chryseis    | 1879, Sept. 11 | Peters        | Clinton
 203. Pompeia     | 1879, Sept. 25 | Peters        | Clinton
 204. Callisto    | 1879, Oct. 8   | Palisa        | Pola
 205. Martha      | 1879, Oct. 13  | Palisa        | Pola
 206. Hersilia    | 1879, Oct. 13  | Peters        | Clinton
 207. Hedda       | 1879, Oct. 17  | Palisa        | Pola
 208. Lachrymosa  | 1879, Oct. 21  | Palisa        | Pola
 209. Dido        | 1879, Oct. 22  | Peters        | Clinton
 210. Isabella    | 1879, Nov. 12  | Palisa        | Pola
 211. Isolda      | 1879, Dec. 10  | Palisa        | Pola
 212. Medea       | 1880, Feb. 6   | Palisa        | Pola
 213. Lilaea      | 1880, Feb. 16  | Peters        | Clinton
 214. Aschera     | 1880, Feb. 26  | Palisa        | Pola
 215. Oenone      | 1880, Apr. 7   | Knorre        | Berlin
 216. Cleopatra   | 1880, Apr. 10  | Palisa        | Pola
 217. Eudora      | 1880, Aug. 30  | Coggia        | Marseilles
 218. Bianca      | 1880, Sept. 4  | Palisa        | Pola
 219. Thusnelda   | 1880, Sept. 20 | Palisa        | Pola
 220. Stephania   | 1881, May 19   | Palisa        | Vienna
 221. Eos         | 1882, Jan. 18  | Palisa        | Vienna
 222. Lucia       | 1882, Feb. 9   | Palisa        | Vienna
 223. Rosa        | 1882, Mar. 9   | Palisa        | Vienna
 224. Oceana      | 1882, Mar. 30  | Palisa        | Vienna
 225. Henrietta   | 1882, Apr. 19  | Palisa        | Vienna
 226. Weringia    | 1882, July 19  | Palisa        | Vienna
 227. Philosophia | 1882, Aug. 12  | Paul Henry    | Paris
 228. Agathe      | 1882, Aug. 19  | Palisa        | Vienna
 229. Adelinda    | 1882, Aug. 22  | Palisa        | Vienna
 230. Athamantis  | 1882, Sept. 3  | De Ball       | Bothcamp
 231. Vindobona   | 1882, Sept. 10 | Palisa        | Vienna
 232. Russia      | 1883, Jan. 31  | Palisa        | Vienna
 233. Asterope    | 1883, May 11   | Borelly       | Marseilles
 234. Barbara     | 1883, Aug. 13  | Peters        | Clinton
 235. Caroline    | 1883, Nov. 29  | Palisa        | Vienna
 236. Honoria     | 1884, Apr. 26  | Palisa        | Vienna
 237. Coelestina  | 1884, June 27  | Palisa        | Vienna
 238. Hypatia     | 1884, July 1   | Knorre        | Berlin
 239. Adrastea    | 1884, Aug. 18  | Palisa        | Vienna
 240. Vanadis     | 1884, Aug. 27  | Borelly       | Marseilles
 241. Germania    | 1884, Sept. 12 | Luther        | Dusseldorf
 242. Kriemhild   | 1884, Sept. 22 | Palisa        | Vienna
 243. Ida         | 1884, Sept. 29 | Palisa        | Vienna
 244. Sita        | 1884, Oct. 14  | Palisa        | Vienna
 245. Vera        | 1885, Feb. 6   | Pogson        | Madras
 246. Asporina    | 1885, Mar. 6   | Borelly       | Marseilles
 247. Eukrate     | 1885, Mar. 14  | Luther        | Dusseldorf
 248. Lameia      | 1885, June 5   | Palisa        | Vienna
 249. Ilse        | 1885, Aug. 17  | Peters        | Clinton
 250. Bettina     | 1885, Sept. 3  | Palisa        | Vienna
 251. Sophia      | 1885, Oct. 4   | Palisa        | Vienna
 252. Clementina  | 1885, Oct. 27  | Perrotin      | Nice
 253. Mathilde    | 1885, Nov. 12  | Palisa        | Vienna
 254. Augusta     | 1886, Mar. 31  | Palisa        | Vienna
 255. Oppavia     | 1886, Mar. 31  | Palisa        | Vienna
 256. Walpurga    | 1886, Apr. 3   | Palisa        | Vienna
 257. Silesia     | 1886, Apr. 5   | Palisa        | Vienna
 258. Tyche       | 1886, May 4    | Luther        | Dusseldorf
 259. Aletheia    | 1886, June 28  | Peters        | Clinton
 260. Huberta     | 1886, Oct. 3   | Palisa        | Vienna
 261. Prymno      | 1886, Oct. 31  | Peters        | Clinton
 262. Valda       | 1886, Nov. 3   | Palisa        | Vienna
 263. Dresda      | 1886, Nov. 3   | Palisa        | Vienna
 264. Libussa     | 1886, Dec. 17  | Peters        | Clinton
 265. Anna        | 1887, Feb. 25  | Palisa        | Vienna
 266. Aline       | 1887, May 17   | Palisa        | Vienna
 267. Tirza       | 1887, May 27   | Charlois      | Nice
 268.             | 1887, June 9   | Borelly       | Marseilles
 269.             | 1887, Sept. 21 | Palisa        | Vienna
 270.             | 1887, Oct. 8   | Peters        | Clinton
 271.             | 1887, Oct. 16  | Knorre        | Berlin
 -----------------+----------------+---------------+------------


3. Remarks on Table I.

The numbers discovered by the thirty-five observers are respectively as
follows:

 Palisa                         60
 Peters                         47
 Luther                         23
 Watson                         22
 Borelly                        15
 Goldschmidt                    14
 Hind                           10
 De Gasparis                     9
 Pogson                          8
 Paul Henry                      7
 Prosper Henry                   7
 Chacornac                       6
 Perrotin                        6
 Coggia                          5
 Knorre                          4
 Tempel                          4
 Ferguson                        3
 Olbers                          2
 Hencke                          2
 Tuttle                          2
 Foerster (with Lesser)          1
 Safford (with Peters)           1
   and Messrs. Charlois,
     Cottenot,
     D'Arrest,
     De Ball,
     Graham,
     Harding,
     Laurent,
     Piazzi,
     Schiaparelli,
     Schulhof,
     Stephan,
     Searle,
     and Tietjen, each          1

Before arrangements had been made for the telegraphic transmission
of discoveries between Europe and America, or even between the
observatories of Europe, the same planet was sometimes independently
discovered by different observers. For example, Virginia was found by
Ferguson, at Washington, on October 4, 1857, and by Luther, at Bilk,
fifteen days later. In all cases, however, credit has been given to the
first observer.

Hersilia, the two hundred and sixth of the group, was lost before
sufficient observations were obtained for determining its elements. It
was not rediscovered till December 14, 1884. Menippe, the one hundred
and eighty-eighth, was also lost soon after its discovery in 1878. It
has not been seen for more than nine years, and considerable uncertainty
attaches to its estimated elements.

Of the two hundred and seventy-one members now known (1887), one hundred
and ninety-one have been discovered in Europe, seventy-four in America,
and six in Asia. The years of most successful search, together with the
number discovered in each, were:

           Asteroids.
 1879         20
 1875         17
 1868         12
 1878         12

And six has been the average yearly number since the commencement of
renewed effort in 1845. All the larger members of the group have,
doubtless, been discovered. It seems not improbable, however, that an
indefinite number of very small bodies belonging to the zone remain to
be found. The process of discovery is becoming more difficult as the
known number increases. The astronomer, for instance, who may discover
number two hundred and seventy-two must know the simultaneous positions
of the two hundred and seventy-one previously detected before he can
decide whether he has picked up a new planet or merely rediscovered an
old one. The numbers discovered in the several months are as follows:

 January      13
 February     23
 March        19
 April        35
 May          21
 June         13
 July         14
 August       28
 September    46
 October      28
 November     26
 December      5

This obvious disparity is readily explained. The weather is favorable
for night watching in April and September; the winter months are too
cold for continuous observations; and the small numbers in June and July
may be referred to the shortness of the nights.


4. Mode of Discovery.

The astronomer who would undertake the search for new asteroids must
supply himself with star-charts extending some considerable distance on
each side of the ecliptic, and containing all telescopic stars down to
the thirteenth or fourteenth magnitude. The detection of a star not
found in the chart of a particular section will indicate its motion, and
hence its planetary character. The construction of such charts has been
a principal object in the labors of Dr. Peters, at Clinton, New York. In
fact, his discovery of minor planets has in most instances been merely
an incidental result of his larger and more important work.


NAMES AND SYMBOLS.

The fact that the names of female deities in the Greek and Roman
mythologies had been given to the first asteroids suggested a similar
course in the selection of names after the new epoch of discovery in
1845. While conformity to this rule has been the general aim of
discoverers, the departures from it have been increasingly numerous. The
twelfth asteroid, discovered in London, was named Victoria, in honor of
the reigning sovereign; the twentieth and twenty-fifth, detected at
Marseilles,[2] received names indicative of the place of their
discovery; Lutetia, the first found at Paris, received its name for a
similar purpose; the fifty-fourth was named Alexandra, for Alexander von
Humboldt; the sixty-seventh, found by Pogson at Madras, was named Asia,
to commemorate the fact that it was the first discovered on that
continent. We find, also, Julia, Bertha, Xantippe, Zelia, Maria,
Isabella, Martha, Dido, Cleopatra, Barbara, Ida, Augusta, and Anna. Why
these were selected we will not stop to inquire.

As the number of asteroids increased it was found inconvenient to
designate them individually by particular signs, as in the case of the
old planets. In 1849, Dr. B. A. Gould proposed to represent them by the
numbers expressing their order of discovery enclosed in a small circle.
This method was at once very generally adopted.


5. Magnitudes of the Asteroids.

The apparent diameter of the largest is less than one-second of arc.
They are all too small, therefore, to be accurately measured by
astronomical instruments. From photometric observations, however,
Argelander,[3] Stone,[4] and Pickering[5] have formed estimates of the
diameters, the results giving probably close approximations to the true
magnitudes. According to these estimates the diameter of the largest,
Vesta, is about three hundred miles, that of Ceres about two hundred,
and those of Pallas and Juno between one and two hundred. The diameters
of about thirty are between fifty and one hundred miles, and those of
all others less than fifty; the estimates for Menippe and Eva giving
twelve and thirteen miles respectively. The diameter of the former is to
that of the earth as one to six hundred and sixty-four; and since
spheres are to each other as the cubes of their diameters, it would
require two hundred and ninety millions of such asteroids to form a
planet as large as our globe. In other words, if the earth be
represented by a sphere one foot in diameter, the magnitude of Menippe
on the same scale would be that of a sand particle whose diameter is one
fifty-fifth of an inch. Its surface contains about four hundred and
forty square miles,--an area equal to a county twenty-one miles square.
The surface attractions of two planets having the same density are to
each other as their diameters. A body, therefore, weighing two hundred
pounds at the earth's surface would on the surface of the asteroid weigh
less than five ounces. At the earth's surface a weight falls sixteen
feet the first second, at the surface of Menippe it would fall about
one-fourth of an inch. A person might leap from its surface to a height
of several hundred feet, in which case he could not return in much less
than an hour. "But of such speculations," Sir John Herschel remarks,
"there is no end."

The number of these planetules between the orbits of Mars and Jupiter in
all probability can never be known. It was estimated by Leverrier that
the quantity of matter contained in the group could not be greater than
one-fourth of the earth's mass. But this would be equal to five thousand
planets, each as large as Vesta, to seventy-two millions as large as
Menippe, or to four thousand millions of five miles in diameter. In
short, the existence of an indefinite number too small for detection by
the most powerful glasses is by no means improbable. The more we study
this wonderful section of the solar system, the more mystery seems to
envelop its origin and constitution.


6. The Orbits of the Asteroids.

The form, magnitude, and position of a planet's orbit are determined by
the following elements:

1. The semi-axis major, or mean distance, denoted by the symbol _a_.

2. The eccentricity, _e_.

3. The longitude of the perihelion, [pi].

4. The longitude of the ascending node, [AN].

5. The inclination, or the angle contained between the plane of the
orbit and that of the ecliptic, _i_.

And in order to compute a planet's place in its orbit for any given time
we must also know

6. Its period, _P_, and

7. Its mean longitude, _l_, at a given epoch.

These elements, except the last, are given for all the asteroids, so far
as known, in Table II. In column first the number denoting the order of
discovery is attached to each name.


TABLE II.

_Elements of the Asteroids._

 -----------------+--------+---------+--------+----------+----------+--------
    Name          |  _a_   |   _P_   |  _e_   |  [pi]    |  [AN]    |   _i_
 -----------------+--------+---------+--------+----------+----------+--------
 149. Medusa      | 2.1327 | 1137.7d | 0.1194 | 246D 37' | 342D 13' |  1D  6'
 244. Sita        | 2.1765 | 1172.8  | 0.1370 |  13   8  | 208  37  |  2  50
 228. Agathe      | 2.2009 | 1192.6  | 0.2405 | 329  23  | 313  18  |  2  33
   8. Flora       | 2.2014 | 1193.3  | 0.1567 |  32  54  | 110  18  |  5  53
  43. Ariadne     | 2.2033 | 1194.5  | 0.1671 | 277  58  | 264  35  |  3  28
 254. Augusta     | 2.2060 | 1196.8  | 0.1227 | 260  47  |  28   9  |  4  36
  72. Feronia     | 2.2661 | 1246.0  | 0.1198 | 307  58  | 207  49  |  5  24
  40. Harmonia    | 2.2673 | 1247.0  | 0.0466 |   0  54  |  93  35  |  4  16
 207. Hedda       | 2.2839 | 1260.7  | 0.0301 | 217   2  |  28  51  |  3  49
 136. Austria     | 2.2863 | 1262.7  | 0.0849 | 316   6  | 186   7  |  9  33
  18. Melpomene   | 2.2956 | 1270.4  | 0.2177 |  15   6  | 150   4  | 10   9
  80. Sappho      | 2.2962 | 1270.9  | 0.2001 | 355  18  | 218  44  |  8  37
 261. Prymno      | 2.3062 | 1278.4  | 0.0794 | 179  35  |  96  33  |  3  38
  12. Victoria    | 2.3342 | 1302.7  | 0.2189 | 301  39  | 235  35  |  8  23
  27. Euterpe     | 2.3472 | 1313.5  | 0.1739 |  87  59  |  93  51  |  1  36
 219. Thusnelda   | 2.3542 | 1319.4  | 0.2247 | 340  34  | 200  44  | 10  47
 163. Erigone     | 2.3560 | 1320.9  | 0.1567 |  93  46  | 159   2  |  4  42
 169. Zelia       | 2.3577 | 1322.3  | 0.1313 | 326  20  | 354  38  |  5  31
   4. Vesta       | 2.3616 | 1325.6  | 0.0884 | 250  57  | 103  29  |  7   8
 186. Celuta      | 2.3623 | 1326.2  | 0.1512 | 327  24  |  14  34  | 13   6
  84. Clio        | 2.3629 | 1326.7  | 0.2360 | 339  20  | 327  28  |  9  22
  51. Nemausa     | 2.3652 | 1328.6  | 0.0672 | 174  43  | 175  52  |  9  57
 220. Stephania   | 2.3666 | 1329.8  | 0.2653 | 332  53  | 258  24  |  7  35
  30. Urania      | 2.3667 | 1329.9  | 0.1266 |  31  46  | 308  12  |  2   6
 105. Artemis     | 2.3744 | 1336.4  | 0.1749 | 242  38  | 188   3  | 21  31
 113. Amalthea    | 2.3761 | 1337.8  | 0.0874 | 198  44  | 123  11  |  5   2
 115. Thyra       | 2.3791 | 1340.3  | 0.1939 |  43   2  | 309   5  | 11  35
 161. Athor       | 2.3792 | 1340.5  | 0.1389 | 310  40  |  18  27  |  9   3
 172. Baucis      | 2.3794 | 1340.6  | 0.1139 | 329  23  | 331  50  | 10   2
 249. Ilse        | 2.3795 | 1340.6  | 0.2195 |  14  17  | 334  49  |  9  40
 230. Athamantis  | 2.3842 | 1344.6  | 0.0615 |  17  31  | 239  33  |  9  26
   7. Iris        | 2.3862 | 1346.4  | 0.2308 |  41  23  | 259  48  |  5  28
   9. Metis       | 2.3866 | 1346.7  | 0.1233 |  71   4  |  68  32  |  5  36
 234. Barbara     | 2.3873 | 1347.3  | 0.2440 | 333  26  | 144   9  | 15  22
  60. Echo        | 2.3934 | 1352.4  | 0.1838 |  98  36  | 192   5  |  3  35
  63. Ausonia     | 2.3979 | 1356.3  | 0.1239 | 270  25  | 337  58  |  5  48
  25. Phocea      | 2.4005 | 1358.5  | 0.2553 | 302  48  | 208  27  | 21  35
 192. Nausicaa    | 2.4014 | 1359.3  | 0.2413 | 343  19  | 160  46  |  6  50
  20. Massalia    | 2.4024 | 1365.8  | 0.1429 |  99   7  | 206  36  |  0  41
 265. Anna        | 2.4096 | 1366.2  | 0.2628 | 226  18  | 335  26  | 25  24
 182. Elsa        | 2.4157 | 1371.4  | 0.1852 |  51  52  | 106  30  |  2   0
 142. Polana      | 2.4194 | 1374.5  | 0.1322 | 219  54  | 317  34  |  2  14
  67. Asia        | 2.4204 | 1375.4  | 0.1866 | 306  35  | 202  47  |  5  59
  44. Nysa        | 2.4223 | 1377.0  | 0.1507 | 111  57  | 131  11  |  3  42
   6. Hebe        | 2.4254 | 1379.3  | 0.2034 |  15  16  | 138  43  | 10  47
  83. Beatrix     | 2.4301 | 1383.6  | 0.0859 | 191  46  |  27  32  |  5   0
 135. Hertha      | 2.4303 | 1383.8  | 0.2037 | 320  11  | 344   3  |  2  19
 131. Vala        | 2.4318 | 1385.1  | 0.0683 | 222  50  |  65  15  |  4  58
 112. Iphigenia   | 2.4335 | 1386.6  | 0.1282 | 338   9  | 324   3  |  2  37
  21. Lutetia     | 2.4354 | 1388.2  | 0.1621 | 327   4  |  80  28  |  3   5
 118. Peitho      | 2.4384 | 1390.8  | 0.1608 |  77  36  |  47  30  |  7  48
 126. Velleda     | 2.4399 | 1392.1  | 0.1061 | 347  46  |  23   7  |  2  56
  42. Isis        | 2.4401 | 1392.2  | 0.2256 | 317  58  |  84  28  |  8  35
  19. Fortuna     | 2.4415 | 1394.4  | 0.1594 |  31   3  | 211  27  |  1  33
  79. Eurynome    | 2.4436 | 1395.2  | 0.1945 |  44  22  | 206  44  |  4  37
 138. Tolosa      | 2.4492 | 1400.0  | 0.1623 | 311  39  |  54  52  |  3  14
 189. Phthia      | 2.4505 | 1401.1  | 0.0356 |   6  50  | 203  22  |  5  10
  11. Parthenope  | 2.4529 | 1403.2  | 0.0994 | 318   2  | 125  11  |  4  37
 178. Belisana    | 2.4583 | 1407.8  | 0.1266 | 278   0  |  50  17  |  2   5
 198. Ampella     | 2.4595 | 1408.9  | 0.2266 | 354  46  | 268  45  |  9  20
 248. Lameia      | 2.4714 | 1419.1  | 0.0656 | 248  40  | 246  34  |  4   1
  17. Thetis      | 2.4726 | 1420.1  | 0.1293 | 261  37  | 125  24  |  5  36
  46. Hestia      | 2.5265 | 1466.8  | 0.1642 | 354  14  | 181  31  |  2  17
  89. Julia       | 2.5510 | 1488.2  | 0.1805 | 353  13  | 311  42  | 16  11
 232. Russia      | 2.5522 | 1489.3  | 0.1754 | 200  25  | 152  30  |  6   4
  29. Amphitrite  | 2.5545 | 1491.3  | 0.0742 |  56  23  | 356  41  |  6   7
 170. Maria       | 2.5549 | 1491.7  | 0.0639 |  95  47  | 301  20  | 14  23
 262. Valda       | 2.5635 | 1496.4  | 0.2172 |  61  42  |  38  40  |  7  46
 258. Tyche       | 2.5643 | 1499.8  | 0.1966 |  15  42  | 208   4  | 14  50
 134. Sophrosyne  | 2.5647 | 1500.3  | 0.1165 |  67  33  | 346  22  | 11  36
 264. Libussa     | 2.5672 | 1502.4  | 0.0925 |   0   7  |  50  23  | 10  29
 193. Ambrosia    | 2.5758 | 1510.0  | 0.2854 |  70  52  | 351  15  | 11  39
  13. Egeria      | 2.5765 | 1510.6  | 0.0871 | 120  10  |  43  12  | 16  32
   5. Astraea     | 2.5786 | 1512.4  | 0.1863 | 134  57  | 141  28  |  5  19
 119. Althea      | 2.5824 | 1515.7  | 0.0815 |  11  29  | 203  57  |  5  45
 157. Dejanira    | 2.5828 | 1516.1  | 0.2105 | 107  24  |  62  31  | 12   2
 101. Helena      | 2.5849 | 1518.0  | 0.1386 | 327  15  | 343  46  | 10  11
  32. Pomona      | 2.5873 | 1520.1  | 0.0830 | 193  22  | 220  43  |  5  29
  91. Aegina      | 2.5895 | 1522.1  | 0.1087 |  80  22  |  11   7  |  2   8
  14. Irene       | 2.5896 | 1522.1  | 0.1627 | 180  19  |  86  48  |  9   8
 111. Ate         | 2.5927 | 1524.8  | 0.1053 | 108  42  | 306  13  |  4  57
 151. Abundantia  | 2.5932 | 1525.3  | 0.0356 | 173  55  |  38  48  |  6  30
  56. Melete      | 2.6010 | 1532.2  | 0.2340 | 294  50  | 194   1  |  8   2
 132. Aethra      | 2.6025 | 1533.5  | 0.3799 | 152  24  | 260   2  | 25   0
 214. Aschera     | 2.6111 | 1541.1  | 0.0316 | 115  55  | 342  30  |  3  27
  70. Panopea     | 2.6139 | 1543.6  | 0.1826 | 299  49  |  48  18  | 11  38
 194. Procne      | 2.6159 | 1545.4  | 0.2383 | 319  33  | 159  19  | 18  24
  53. Calypso     | 2.6175 | 1546.8  | 0.2060 |  92  52  | 143  58  |  5   7
  78. Diana       | 2.6194 | 1548.5  | 0.2088 | 121  42  | 333  58  |  8  40
 124. Alceste     | 2.6297 | 1557.6  | 0.0784 | 245  42  | 188  26  |  2  56
  23. Thalia      | 2.6306 | 1558.4  | 0.2299 | 123  58  |  67  45  | 10  14
 164. Eva         | 2.6314 | 1559.1  | 0.3471 | 359  32  |  77  28  | 24  25
  15. Eunomia     | 2.6437 | 1570.0  | 0.1872 |  27  52  | 188  26  |  2  56
  37. Fides       | 2.6440 | 1570.3  | 0.1758 |  66  26  |   8  21  |  3   7
  66. Maia        | 2.6454 | 1571.6  | 0.1750 |  48   8  |   8  17  |  3   6
 224. Oceana      | 2.6465 | 1572.6  | 0.0455 | 270  51  | 353  18  |  5  52
 253. Mathilde    | 2.6469 | 1572.9  | 0.2620 | 333  39  | 180   3  |  6  37
  50. Virginia    | 2.6520 | 1577.4  | 0.2852 |  10   9  | 173  45  |  2  48
 144. Vibilia     | 2.6530 | 1578.4  | 0.2348 |   7   9  |  76  47  |  4  48
  85. Io          | 2.6539 | 1579.2  | 0.1911 | 322  35  | 203  56  | 11  53
  26. Proserpine  | 2.6561 | 1581.1  | 0.0873 | 236  25  |  45  55  |  3  36
 233. Asterope    | 2.6596 | 1584.3  | 0.1010 | 344  36  | 222  25  |  7  39
 102. Miriam      | 2.6619 | 1586.3  | 0.3035 | 354  39  | 211  58  |  5   4
 240. Vanadis     | 2.6638 | 1588.0  | 0.2056 |  51  53  | 114  54  |  2   6
  73. Clytie      | 2.6652 | 1589.3  | 0.0419 |  57  55  |   7  51  |  2  24
 218. Bianca      | 2.6653 | 1589.3  | 0.1155 | 230  14  | 170  50  | 15  13
 141. Lumen       | 2.6666 | 1590.5  | 0.2115 |  13  43  | 319   7  | 11  57
  77. Frigga      | 2.6680 | 1591.8  | 0.1318 |  58  47  |   2   0  |  2  28
   3. Juno        | 2.6683 | 1592.0  | 0.2579 |  54  50  | 170  53  | 13   1
  97. Clotho      | 2.6708 | 1594.3  | 0.2550 |  65  32  | 160  37  | 11  46
  75. Eurydice    | 2.6720 | 1595.3  | 0.3060 | 335  33  | 359  56  |  5   1
 145. Adeona      | 2.6724 | 1595.4  | 0.1406 | 117  53  |  77  41  | 12  38
 204. Callisto    | 2.6732 | 1596.4  | 0.1752 | 257  45  | 205  40  |  8  19
 114. Cassandra   | 2.6758 | 1598.8  | 0.1401 | 153   6  | 164  24  |  4  55
 201. Penelope    | 2.6764 | 1599.3  | 0.1818 | 334  21  | 157   5  |  5  44
  64. Angelina    | 2.6816 | 1603.9  | 0.1271 | 125  36  | 311   4  |  1  19
  98. Ianthe      | 2.6847 | 1606.7  | 0.1920 | 148  52  | 354   7  | 15  32
  34. Circe       | 2.6864 | 1608.3  | 0.1073 | 148  41  | 184  46  |  5  27
 123. Brunhilda   | 2.6918 | 1613.2  | 0.1150 |  72  57  | 308  28  |  6  27
 166. Rhodope     | 2.6927 | 1613.9  | 0.2140 |  30  51  | 129  33  | 12   2
 109. Felicitas   | 2.6950 | 1616.0  | 0.3002 |  56   1  |   4  56  |  8   3
 246. Asporina    | 2.6994 | 1619.9  | 0.1065 | 255  54  | 162  35  | 15  39
  58. Concordia   | 2.7004 | 1620.8  | 0.0426 | 189  10  | 161  20  |  5   2
 103. Hera        | 2.7014 | 1621.8  | 0.0803 | 321   3  | 136  18  |  5  24
  54. Alexandra   | 2.7095 | 1629.1  | 0.2000 | 295  39  | 313  45  | 11  47
 226. Weringia    | 2.7118 | 1631.2  | 0.2048 | 284  46  | 135  18  | 15  50
  59. Olympia     | 2.7124 | 1631.7  | 0.1189 |  17  33  | 170  26  |  8  37
 146. Lucina      | 2.7189 | 1637.5  | 0.0655 | 227  34  |  84  16  | 13   6
  45. Eugenia     | 2.7205 | 1639.0  | 0.0811 | 232   5  | 147  57  |  6  35
 210. Isabella    | 2.7235 | 1641.7  | 0.1220 |  44  22  |  32  58  |  5  18
 187. Lamberta    | 2.7272 | 1645.0  | 0.2391 | 214   4  |  22  13  | 10  43
 180. Garumna     | 2.7286 | 1646.3  | 0.1722 | 125  56  | 314  42  |  0  54
 160. Una         | 2.7287 | 1646.4  | 0.0624 |  55  57  |   9  22  |  3  51
 140. Siwa        | 2.7316 | 1649.0  | 0.2160 | 300  33  | 107   2  |  3  12
 110. Lydia       | 2.7327 | 1650.0  | 0.0770 | 336  49  |  57  10  |  6   0
 185. Eunice      | 2.7372 | 1654.1  | 0.1292 |  16  32  | 153  50  | 23  17
 203. Pompeia     | 2.7376 | 1654.5  | 0.0588 |  42  51  | 348  37  |  3  13
 200. Dynamene    | 2.7378 | 1654.6  | 0.1335 |  46  38  | 325  26  |  6  56
 197. Arete       | 2.7390 | 1655.8  | 0.1621 | 324  51  |  82   6  |  8  48
 206. Hersilia    | 2.7399 | 1656.5  | 0.0389 |  95  44  | 145  16  |  3  46
 255. Oppavia     | 2.7402 | 1656.6  | 0.0728 | 169  15  |  14   6  |  9  33
 247. Eukrate     | 2.7412 | 1657.7  | 0.2387 |  53  44  |   0  20  | 25   7
  38. Leda        | 2.7432 | 1659.6  | 0.1531 | 101  20  | 296  27  |  6  57
 125. Liberatrix  | 2.7437 | 1660.0  | 0.0798 | 273  29  | 169  35  |  4  38
 173. Ino         | 2.7446 | 1660.8  | 0.2047 |  13  28  | 148  34  | 14  15
  36. Atalanta    | 2.7452 | 1661.3  | 0.3023 |  42  44  | 359  14  | 18  42
 128. Nemesis     | 2.7514 | 1666.9  | 0.1257 |  16  34  |  76  31  |  6  16
  93. Minerva     | 2.7537 | 1669.0  | 0.1405 | 274  44  |   5   4  |  8  37
 127. Johanna     | 2.7550 | 1670.3  | 0.0659 | 122  37  |  31  46  |  8  17
  71. Niobe       | 2.7558 | 1671.0  | 0.1732 | 221  17  | 316  30  | 23  19
 213. Lilaea      | 2.7563 | 1671.4  | 0.1437 | 281   4  | 122  17  |  6  47
  55. Pandora     | 2.7604 | 1675.1  | 0.1429 |  10  36  |  10  56  |  7  14
 237. Coelestina  | 2.7607 | 1675.5  | 0.0738 | 282  49  |  84  33  |  9  46
 143. Adria       | 2.7619 | 1676.6  | 0.0729 | 222  27  | 333  42  | 11  30
  82. Alcmene     | 2.7620 | 1676.6  | 0.2228 | 131  45  |  26  57  |  2  51
 116. Sirona      | 2.7669 | 1681.1  | 0.1433 | 152  47  |  64  26  |  3  35
   1. Ceres       | 2.7673 | 1681.4  | 0.0763 | 149  38  |  80  47  | 10  37
  88. Thisbe      | 2.7673 | 1681.5  | 0.1632 | 308  34  | 277  54  | 16  11
 215. Oenone      | 2.7679 | 1682.0  | 0.0390 | 346  24  |  25  25  |  1  44
   2. Pallas      | 2.7680 | 1682.1  | 0.2408 | 122  12  | 172  45  | 34  44
  39. Laetitia    | 2.7680 | 1682.1  | 0.1142 |   3   8  | 157  15  | 10  22
  41. Daphne      | 2.7688 | 1682.8  | 0.2674 | 220  33  | 179   8  | 15  58
 177. Irma        | 2.7695 | 1683.5  | 0.2370 |  22   6  | 349  17  |  1  27
 148. Gallia      | 2.7710 | 1684.8  | 0.1855 |  36   7  | 145  13  | 25  21
 267. Tirza       | 2.7742 | 1687.6  | 0.0986 | 264   5  |  73  59  |  6   2
  74. Galatea     | 2.7770 | 1690.3  | 0.2392 |   8  18  | 197  51  |  4   0
 205. Martha      | 2.7771 | 1690.4  | 0.1752 |  21  54  | 212  12  | 10  40
 139. Juewa       | 2.7793 | 1692.4  | 0.1773 | 164  34  |   2  21  | 10  57
  28. Bellona     | 2.7797 | 1692.7  | 0.1491 | 124   1  | 144  37  |  9  22
  68. Leto        | 2.7805 | 1693.5  | 0.1883 | 345  14  |  45   1  |  7  58
 216. Cleopatra   | 2.7964 | 1708.0  | 0.2492 | 328  15  | 215  49  | 13   2
  99. Dike        | 2.7966 | 1708.3  | 0.2384 | 240  36  |  41  44  | 13  53
 236. Honoria     | 2.7993 | 1710.7  | 0.1893 | 356  59  | 186  27  |  7  37
 183. Istria      | 2.8024 | 1713.4  | 0.3530 |  45   0  | 142  46  | 26  33
 266. Aline       | 2.8078 | 1718.5  | 0.1573 |  23  52  | 236  18  | 13  20
 188. Menippe     | 2.8211 | 1730.7  | 0.2173 | 309  38  | 241  44  | 11  21
 167. Urda        | 2.8533 | 1760.4  | 0.0340 | 296   4  | 166  28  |  2  11
  81. Terpsichore | 2.8580 | 1764.8  | 0.2080 |  49   1  |   2  25  |  7  55
 174. Phaedra     | 2.8600 | 1766.6  | 0.1492 | 253  12  | 328  49  | 12   9
 243. Ida         | 2.8610 | 1767.5  | 0.0419 |  71  22  | 326  21  |  1  10
 242. Kriemhild   | 2.8623 | 1768.7  | 0.1219 | 123   1  | 207  57  | 11  17
 129. Antigone    | 2.8678 | 1773.9  | 0.2126 | 242   4  | 137  37  | 12  10
 217. Eudora      | 2.8690 | 1774.9  | 0.3068 | 314  41  | 164  10  | 10  19
 158. Coronis     | 2.8714 | 1777.2  | 0.0545 |  56  56  | 281  30  |  1   0
  33. Polyhymnia  | 2.8751 | 1780.7  | 0.3349 | 342  59  |   9  19  |  1  56
 195. Euryclea    | 2.8790 | 1784.2  | 0.0471 | 115  48  |   7  57  |  7   1
 235. Caroline    | 2.8795 | 1784.7  | 0.0595 | 268  29  |  66  35  |  9   4
  47. Aglaia      | 2.8819 | 1786.9  | 0.1317 | 312  40  |  40  20  |  5   1
 208. Lachrymosa  | 2.8926 | 1796.9  | 0.0149 | 127  52  |   5  43  |  1  48
 191. Kolga       | 2.8967 | 1800.8  | 0.0876 |  23  21  | 159  47  | 11  29
  22. Calliope    | 2.9090 | 1801.0  | 0.0193 |  62  43  |   4  47  |  1  45
 155. Scylla      | 2.9127 | 1815.7  | 0.2559 |  82   1  |  42  52  | 14   4
 238. Hypatia     | 2.9163 | 1819.0  | 0.0946 |  32  18  | 184  26  | 12  28
 231. Vindobona   | 2.9192 | 1821.7  | 0.1537 | 253  23  | 352  49  |  5  10
  16. Psyche      | 2.9210 | 1823.4  | 0.1392 |  15   9  | 150  36  |  3   4
 179. Clytemnestra| 2.9711 | 1870.6  | 0.1133 | 355  39  | 253  13  |  7  47
 239. Adrastea    | 2.9736 | 1873.0  | 0.2279 |  26   1  | 181  34  |  6   4
  69. Hesperia    | 2.9779 | 1877.0  | 0.1712 | 108  19  | 187  12  |  8  28
 150. Nuwa        | 2.9785 | 1877.5  | 0.1307 | 355  27  | 207  35  |  2   9
  61. Danae       | 2.9855 | 1884.2  | 0.1615 | 344   4  | 334  11  | 18  14
 117. Lomia       | 2.9907 | 1889.1  | 0.0229 |  48  46  | 349  39  | 14  58
  35. Leucothea   | 2.9923 | 1890.6  | 0.2237 | 202  25  | 355  49  |  8  12
 263. Dresda      | 3.0120 | 1909.3  | 0.3051 | 308  49  | 217  56  |  1  27
 221. Eos         | 3.0134 | 1910.7  | 0.1028 | 330  58  | 142  35  | 10  51
 162. Laurentia   | 3.0241 | 1920.8  | 0.1726 | 145  52  |  38  15  |  6   4
 156. Xantippe    | 3.0375 | 1933.7  | 0.2637 | 155  58  | 246  11  |  7  29
 241. Germania    | 3.0381 | 1934.0  | 0.1013 | 340   7  | 272  28  |  5  30
 256. Walpurga    | 3.0450 | 1940.8  | 0.1180 | 240  17  | 183  35  | 12  44
 211. Isolda      | 3.0464 | 1942.2  | 0.1541 |  74  12  | 265  29  |  3  51
  96. Aegle       | 3.0497 | 1945.3  | 0.1405 | 163  10  | 322  50  | 16   7
 257. Silesia     | 3.0572 | 1952.5  | 0.2555 |  54  16  |  34  31  |  4  41
 133. Cyrene      | 3.0578 | 1953.0  | 0.1398 | 247  13  | 321   8  |  7  14
  95. Arethusa    | 3.0712 | 1965.9  | 0.1447 |  32  58  | 244  17  | 12  54
 202. Chryseis    | 3.0777 | 1972.1  | 0.0959 | 129  46  | 137  47  |  8  48
 268. ----        | 3.0852 | 1973.9  | 0.1285 | 184  48  | 121  53  |  2  25
 100. Hecate      | 3.0904 | 1984.3  | 0.1639 | 308   3  | 128  12  |  6  23
  49. Pales       | 3.0908 | 1984.7  | 0.2330 |  31  15  | 290  40  |  3   8
 223. Rosa        | 3.0940 | 1987.9  | 0.1186 | 102  48  |  49   0  |  1  59
  52. Europa      | 3.0955 | 1988.0  | 0.1098 | 106  57  | 129  40  |  7  27
 245. Vera        | 3.0985 | 1992.1  | 0.1950 |  25  29  |  62  37  |  5  10
  86. Semele      | 3.1015 | 1995.1  | 0.2193 |  29  10  |  87  45  |  4  47
 159. Aemilia     | 3.1089 | 2002.2  | 0.1034 | 101  22  | 135   9  |  6   4
  48. Doris       | 3.1127 | 2005.9  | 0.0649 |  70  33  | 184  55  |  6  31
 196. Philomela   | 3.1137 | 2006.8  | 0.0118 | 309  19  |  73  24  |  7  16
 130. Electra     | 3.1145 | 2007.7  | 0.2132 |  20  34  | 146   6  | 22  57
 212. Medea       | 3.1157 | 2008.8  | 0.1013 |  56  18  | 315  16  |  4  16
 120. Lachesis    | 3.1211 | 2014.0  | 0.0475 | 214   0  | 342  51  |  7   1
 181. Eucharis    | 3.1226 | 2015.4  | 0.2205 |  95  25  | 144  45  | 18  38
  62. Erato       | 3.1241 | 2016.9  | 0.1756 |  39   0  | 125  46  |  2  12
 222. Lucia       | 3.1263 | 2019.0  | 0.1453 | 258   2  |  80  11  |  2  11
 137. Meliboea    | 3.1264 | 2019.1  | 0.2074 | 307  58  | 204  22  | 13  22
 165. Loreley     | 3.1269 | 2019.6  | 0.0734 | 223  50  | 304   6  | 10  12
 251. Sophia      | 3.1315 | 2024.1  | 0.1243 |  77   7  | 157   6  | 10  20
  24. Themis      | 3.1357 | 2028.1  | 0.1242 | 144   8  |  35  49  |  0  49
 152. Atala       | 3.1362 | 2028.6  | 0.0862 |  84  23  |  41  29  | 12  12
  10. Hygeia      | 3.1366 | 2029.1  | 0.1156 | 237   2  | 285  38  |  3  49
 259. Aletheia    | 3.1369 | 2029.3  | 0.1176 | 241  45  |  88  32  | 10  40
 227. Philosophia | 3.1393 | 2031.6  | 0.2131 | 226  23  | 330  52  |  9  16
 147. Protogenea  | 3.1393 | 2031.6  | 0.0247 |  25  38  | 251  16  |  1  54
 171. Ophelia     | 3.1432 | 2035.4  | 0.1168 | 143  59  | 101  10  |  2  34
 209. Dido        | 3.1436 | 2035.9  | 0.0637 | 257  33  |   2   0  |  7  15
  31. Euphrosyne  | 3.1468 | 2039.0  | 0.2228 |  93  26  |  31  31  | 26  27
  90. Antiope     | 3.1475 | 2039.7  | 0.1645 | 301  15  |  71  29  |  2  17
 104. Clymene     | 3.1507 | 2042.7  | 0.1579 |  59  32  |  43  32  |  2  54
  57. Mnemosyne   | 3.1510 | 2043.0  | 0.1145 |  53  25  | 200   2  | 15  12
 250. Bettina     | 3.1524 | 2044.3  | 0.1302 |  87  28  |  26  12  | 12  54
 252. Clementina  | 3.1552 | 2047.1  | 0.0837 | 355   8  | 208  19  | 10   2
  94. Aurora      | 3.1602 | 2052.0  | 0.0827 |  48  46  |   4   9  |  8   4
 106. Dione       | 3.1670 | 2058.6  | 0.1788 |  25  57  |  63  14  |  4  38
 199. Byblis      | 3.1777 | 2069.0  | 0.1687 | 261  20  |  89  52  | 15  22
  92. Undina      | 3.1851 | 2076.3  | 0.1024 | 331  27  | 102  52  |  9  57
 184. Deiopea     | 3.1883 | 2079.4  | 0.0725 | 169  22  | 336  18  |  1  12
 176. Idunna      | 3.1906 | 2081.6  | 0.1641 |  20  34  | 201  13  | 22  31
 154. Bertha      | 3.1976 | 2088.5  | 0.0788 | 190  47  |  37  35  | 20  59
 108. Hecuba      | 3.2113 | 2101.0  | 0.1005 | 173  49  | 352  17  |  4  24
 122. Gerda       | 3.2177 | 2108.2  | 0.0415 | 203  45  | 178  43  |  1  36
 168. Sibylla     | 3.3765 | 2266.2  | 0.0707 |  11  26  | 209  47  |  4  33
 225. Henrietta   | 3.4007 | 2277.8  | 0.2661 | 299  13  | 200  45  | 20  45
 229. Adelinda    | 3.4129 | 2302.9  | 0.1562 | 332   7  |  30  49  |  2  11
  76. Freia       | 3.4140 | 2304.1  | 0.1700 |  90  49  | 212   5  |  2   3
 260. Huberta     | 3.4212 | 2311.5  | 0.1113 | 313  22  | 168  48  |  6  18
  65. Maximiliana | 3.4270 | 2317.2  | 0.1097 | 260  36  | 158  50  |  3  29
 121. Hermione    | 3.4535 | 2344.2  | 0.1255 | 357  50  |  76  46  |  7  36
  87. Sylvia      | 3.4833 | 2374.5  | 0.0922 | 333  48  |  75  49  | 10  55
 107. Camilla     | 3.4847 | 2376.0  | 0.0756 | 115  53  | 176  18  |  9  54
 175. Andromache  | 3.5071 | 2399.0  | 0.3476 | 293   0  |  23  35  |  3  46
 190. Ismene      | 3.9471 | 2864.3  | 0.1634 | 105  39  | 177   0  |  6   7
 153. Hilda       | 3.9523 | 2869.9  | 0.1721 | 285  47  | 228  20  |  7  55
 -----------------+--------+---------+--------+----------+----------+--------




PART II.




DISCUSSION OF THE FACTS IN TABLE II.


1. Extent of the Zone.

In Table II. the unit of column _a_ is the earth's mean distance from
the sun, or ninety-three million miles. On this scale the breadth of the
zone is 1.8196. Or, if we estimate the breadth from the perihelion of
Aethra (1.612) to the aphelion of Andromache (4.726), it is 3.114,--more
than three times the radius of the earth's orbit. A very remarkable
characteristic of the group is the interlacing or intertwining of
orbits. "One fact," says D'Arrest, "seems above all to confirm the idea
of an intimate relation between all the minor planets; it is, that if
their orbits are figured under the form of material rings, these rings
will be found so entangled that it would be possible, by means of one
among them taken at hazard, to lift up all the rest."[6] Our present
knowledge of this wide and complicated cluster is the result of a vast
amount, not only of observations, but also of mathematical labor. In
view, however, of the perturbations of these bodies by the larger
planets, and especially by Jupiter, it is easy to see that the
discussion of their motions must present a field of investigation
practically boundless.

While the known minor planets were but few in number the theory of
Olbers in regard to their origin seemed highly probable; it has,
however, been completely disproved by more recent discoveries. The
breadth of the zone being now greater than the distance of Mars from the
sun, it is no more probable that the asteroids were produced by the
disruption of a single planet than that Mercury, Venus, the earth, and
Mars originated in a similar manner.


2. The Small Mass of the Asteroids.

In taking a general view of the solar system we cannot fail to be struck
by the remarkable fact that Jupiter, whose mass is much greater than
that of all other planets united, should be immediately succeeded by a
region so nearly destitute of matter as the zone of asteroids. Leverrier
inferred from the motion of Mars's perihelion that the mass of Jupiter
is at least twelve hundred times greater than that of all the planets in
the asteroid ring. The fact is suggestive of Jupiter's dominating energy
in the evolution of the asteroid system. We find also something
analogous among the satellites of Jupiter, Saturn, and Uranus. Jupiter's
third satellite, the largest of the number, is nearly four times greater
than the second. Immediately within the orbit of Titan, the largest
satellite of Saturn, occurs a wide hiatus, and the volume of the next
interior satellite is to that of Titan in the ratio of one to
twenty-one. In the Uranian system the widest interval between adjacent
orbits is just within the orbit of the bright satellite, Titania.

The foregoing facts suggest the inquiry, What effect would be produced
by a large planet on interior masses abandoned by a central spheroid? As
the phenomena in all instances would be of the same nature, we will
consider a single case,--that of Jupiter and the asteroids.

The powerful mass of the exterior body would produce great perturbations
of the neighboring small planets abandoned at the solar equator. The
disturbed orbits, in some cases, would thus attain considerable
eccentricity, so that the matter moving in them would, in perihelion, be
brought in contact with the equatorial parts of the central body, and
thus become reunited with it.[7] The extreme rarity of the zone between
Mars and Jupiter, regarded as a single ring, is thus accounted for in
accordance with known dynamical laws.


3. The Limits of Perihelion Distance.

It is sufficiently obvious that whenever the perihelion distance of a
planet or comet is less than the sun's radius, a collision must occur as
the moving body approaches the focus of its path. The great comet of
1843 passed so near the sun as almost to graze its surface. With a
perihelion distance but very slightly less, it would have been
precipitated into the sun and incorporated with its mass. In former
epochs, when the dimensions of the sun were much greater than at
present, this falling of comets into the central orb of the system must
have been a comparatively frequent occurrence. Again, if Mercury's orbit
had its present eccentricity when the radius of the solar spheroid was
twenty-nine million miles, the planet at its nearest approach to the
centre of its motion must have passed through the outer strata of the
central body. In such case a lessening of the planet's mean distance
would be a necessary consequence. We thus see that in the formation of
the solar system the eccentricity of an asteroidal orbit could not
increase beyond a moderate limit without the planet's return to the
solar mass. The bearing of these views on the arrangement of the minor
planets will appear in what follows.


4. Was the Asteroid Zone originally Stable?--Distribution of the Members
in Space.

One of the most interesting discoveries of the eighteenth century was
Lagrange's law securing the stability of the solar system. This
celebrated theorem, however, is not to be understood in an absolute or
unlimited sense. It makes no provision against the effect of a resisting
medium, or against the entrance of cosmic matter from without. It does
not secure the stability of all periodic comets nor of the meteor
streams revolving about the sun. In the early stages of the system's
development the matter moving in unstable orbits may have been, and
probably was, much more abundant than at present. But even now, are we
justified in concluding that all known asteroids have stable orbits? For
the major planets the secular variations of eccentricity have been
calculated, but for the orbits between Mars and Jupiter these limits are
unknown. With an eccentricity of 0.252 (less than that of many
asteroids), the distance of Hilda's aphelion would be greater than that
of Jupiter's perihelion. It seems possible, therefore, that certain
minor planets may have their orbits much changed by Jupiter's disturbing
influence.[8]

Whoever looks at a table of asteroids arranged in their order of
discovery will find only a perplexing mass of figures. Whether we regard
their distances, their inclinations, or the forms of their orbits, the
elements of the members are without any obvious connection. Nor is the
confusion lessened when the orbits are drawn and presented to the eye.
In fact, the crossing and recrossing of so many ellipses of various
forms merely increase the entanglement. But can no order be traced in
all this complexity? Are there no breaks or vacant spaces within the
zone's extreme limits? Has Jupiter's influence been effective in fixing
the position and arrangement of the cluster? Such are some of the
questions demanding our attention. If "the universe is a book written
for man's reading," patient study may resolve the problem contained in
these mysterious leaves.

Simultaneously with the discovery of new members in the cluster of minor
planets, near the middle of the century, occurred the resolution of the
great nebula in Orion. This startling achievement by Lord Rosse's
telescope was the signal for the abandonment of the nebular hypothesis
by many of its former advocates. To the present writer, however, the
partial resolution of a single nebula seemed hardly a sufficient reason
for its summary rejection. The question then arose whether any probable
test of Laplace's theory could be found in the solar system itself. The
train of thought was somewhat as follows: Several new members have been
found in the zone of asteroids; its dimensions have been greatly
extended, so that we can now assign no definite limits either to the
ring itself or to the number of its planets; if the nebular hypothesis
be true, the sun, after Jupiter's separation, extended successively to
the various decreasing distances of the several asteroids; the
eccentricities of these bodies are generally greater than those of the
old planets; this difference is probably due to the disturbing force of
Jupiter; the zone includes several distances at which the periods of
asteroids would be commensurable with that of Jupiter; in such case the
conjunctions of the minor with the major planet would occur in the same
parts of its path, the disturbing effects would accumulate, and the
eccentricity would become very marked; such bodies in perihelion would
return to the sun, and hence blanks or chasms would be formed in
particular parts of the zone. On the other hand, if the nebular
hypothesis was not true, the occurrence of these gaps was not to be
expected. Having thus pointed out a prospective test of the theory, it
was announced with some hesitation that _those parts of the asteroid
zone in which a simple relation of commensurability would obtain between
the period of a minor planet and that of Jupiter are distinguished as
gaps or chasms similar to the interval in Saturn's ring_.

The existence of these blanks was thus predicted in theory before it was
established as a fact of observation. When the law was first publicly
stated in 1866, but ten asteroids had been found with distances greater
than three times that of the earth. The number of such now known is
sixty-five. For more than a score of years the progress of discovery
has been watched with lively interest, and the one hundred and eighty
new members of the group have been found moving in harmony with this law
of distribution.[9]


COMMENSURABILITY OF PERIODS.

When we say that an asteroid's period is commensurable with that of
Jupiter, we mean that a certain whole number of the former is equal to
another whole number of the latter. For instance, if a minor planet
completes two revolutions to Jupiter's one, or five to Jupiter's two,
the periods are commensurable. It must be remarked, however, that
Jupiter's effectiveness in disturbing the motion of a minor planet
depends on the _order_ of commensurability. Thus, if the ratio of the
less to the greater period is expressed by the fraction 1/2, where the
difference between the numerator and the denominator is one, the
commensurability is of the first order; 1/3 is of the second; 2/5, of
the third, etc. The difference between the terms of the ratio indicates
the frequency of conjunctions while Jupiter is completing the number
of revolutions expressed by the numerator. The distance 3.277,
corresponding to the ratio 1/2, is the only case of the first order in
the entire ring; those of the second order, answering to 1/3 and 3/5,
are 2.50 and 3.70. These orders of commensurability may be thus arranged
in a tabular form, the radius of the earth's orbit being the unit of
distance:

 +--------+----------------+-----------+
 | Order. |     Ratio.     | Distance. |
 +--------+----------------+-----------+
 | First  |      1/2       |   3.277   |
 |        |                |           |
 | Second |    1/3, 3/5    | { 2.50    |
 |        |                | { 3.70    |
 |        |                |           |
 |        |                | { 2.82    |
 | Third  |  2/5, 4/7, 5/8 | { 3.58    |
 |        |                | { 3.80    |
 |        |                |           |
 |        |                | { 2.95    |
 | Fourth | 3/7, 5/9, 7/11 | { 3.51    |
 |        |                | { 3.85    |
 +--------+----------------+-----------+

Do these parts of the ring present discontinuities? and, if so, can they
be ascribed to a chance distribution? Let us consider them in order.


I.--The Distance 3.277.

At this distance an asteroid's conjunctions with Jupiter would all occur
at the same place, and its perturbations would be there repeated at
intervals equal to Jupiter's period (11.86 y.). Now, when the asteroids
are arranged in the order of their mean distances (as in Table II.) this
part of the zone presents a wide chasm. The space between 3.218 and
3.376 remains, hitherto a perfect blank, while the adjacent portions of
equal breadth, interior and exterior, contain fifty-four minor planets.
The probability that this distribution is not the result of chance is
more than three hundred billions to one.

The breadth of this chasm is one-twentieth part of its distance from the
sun, or one-eleventh part of the breadth of the entire zone.


II.--The Second Order of Commensurability.--The Distances 2.50 and 3.70.

At the former of these distances an asteroid's period would be one-third
of Jupiter's, and at the latter, three-fifths. That part of the zone
included between the distances 2.30 and 2.70 contains one hundred and
ten intervals, exclusive of the maximum at the critical distance 2.50.
This gap--between Thetis and Hestia--is not only much greater than any
other of this number, but is more than sixteen times greater than their
average. The distance 3.70 falls in the wide hiatus interior to the
orbit of Ismene.


III.--Chasms corresponding to the Third Order.--The Distances 2.82,
3.58, and 3.80.

As the order of commensurability becomes less simple, the corresponding
breaks in the zone are less distinctly marked. In the present case
conjunctions with Jupiter would occur at angular intervals of 120D. The
gaps, however, are still easily perceptible. Between the distances 2.765
and 2.808 we find twenty minor planets. In the next exterior space of
equal breadth, containing the distance 2.82, there is but one. This is
No. 188, Menippe, whose elements are still somewhat uncertain. The space
between 2.851 and 2.894--that is, the part of equal extent immediately
beyond the gap--contains thirteen asteroids. The distances 3.58 and 3.80
are in the chasm between Andromache and Ismene.


IV.--The Distances 2.95, 3.51,[10] and 3.85, corresponding to the Fourth
Order of Commensurability.

The first of these distances is in the interval between Psyche and
Clytemnestra; the second and third, in that exterior to Andromache.

The nine cases considered are the only ones in which the conjunctions
with Jupiter would occur at less than five points of an asteroid's
orbit. Higher orders of commensurability may perhaps be neglected. It
will be seen, however, that the distances 2.25, 2.70, 3.03, and 3.23,
corresponding to the ratios of the fifth order, 2/7, 3/8, 4/9, and 6/11,
still afford traces of Jupiter's influence. The first is in the interval
between Augusta and Feronia; the last falls in the same gap with 3.277;
and the second and third are in breaks less distinctly marked. It may
also be worthy of notice that the rather wide interval between Prymno
and Victoria is where ten periods of a minor planet would be equal to
three of Jupiter. The distance of Medusa is somewhat uncertain.

The FACT of the existence of well-defined gaps in the designated parts
of the ring has been clearly established. But the theory of probability
applied in a single instance gives, as we have seen, but one chance in
300,000,000,000 that the distribution is accidental. This improbability
is increased many millions of times when we include all the gaps
corresponding to simple cases of commensurability. We conclude,
therefore, that those discontinuities cannot be referred to a chance
arrangement. What, then, was their physical cause? and what has become
of the eliminated asteroids?

What was said in regard to the limits of perihelion distance may suggest
a possible answer to these interesting questions. The doctrine of the
sun's gradual contraction is now accepted by a majority of astronomers.
According to this theory the solar radius at an epoch not relatively
remote was twice what it is at present. At anterior stages it was 0.4,
1.0, 2.0,[11] etc. At the first mentioned the comets of 1843 and 1668,
as well as several others, could not have been moving in their present
orbits, since in perihelion they must have plunged into the sun. At the
second, Encke's comet and all others with perihelia within Mercury's
orbit would have shared a similar fate. At the last named all asteroids
with perihelion distances less than two would have been re-incorporated
with the central mass. As the least distance of Aethra is but 1.587, its
orbit could not have had its present form and dimensions when the radius
of the solar nebula was equal to the aphelion distance of Mars (1.665).

It is easy to see, therefore, that in those parts of the ring where
Jupiter would produce extraordinary disturbance the formation of chasms
would be very highly probable.


5. Relations between certain Adjacent Orbits.

The distances, periods, inclinations, and eccentricities of Hilda and
Ismene, the outermost pair of the group, are very nearly identical. It
is a remarkable fact, however, that the longitudes of their perihelia
differ by almost exactly 180D. Did they separate at nearly the same
time from opposite sides of the solar nebula? Other adjacent pairs
having a striking similarity between their orbital elements are Sirona
and Ceres, Fides and Maia, Fortuna and Eurynome, and perhaps a few
others. Such coincidences can hardly be accidental. Original asteroids,
soon after their detachment from the central body, may have been
separated by the sun's unequal attraction on their parts. Such divisions
have occurred in the world of comets, why not also in the cluster of
minor planets?


6. The Eccentricities.

The least eccentric orbit in the group is that of Philomela (196); the
most eccentric that of Aethra (132). Comparing these with the orbit of
the second comet of 1867 we have

 The eccentricity of                     Philomela = 0.01
  "        "      "                          Aethra = 0.38
  "        "      "  Comet II. 1867 (ret. in 1885) = 0.41

The orbit of Aethra, it is seen, more nearly resembles the last than the
first. It might perhaps be called the connecting-link between planetary
and cometary orbits.

The average eccentricity of the two hundred and sixty-eight asteroids
whose orbits have been calculated is 0.1569. As with the orbits of the
old planets, the eccentricities vary within moderate limits, some
increasing, others diminishing. The average, however, will probably
remain very nearly the same. An inspection of the table shows that while
but one orbit is less eccentric than the earth's, sixty-nine depart more
from the circular form than the orbit of Mercury. These eccentricities
seem to indicate that the forms of the asteroidal orbits were influenced
by special causes. It may be worthy of remark that the eccentricity does
not appear to vary with the distance from the sun, being nearly the same
for the interior members of the zone as for the exterior.


7. The Inclinations.

The inclinations in Table II. are thus distributed:

 From 0D to  4D    70
  "   4D to  8D    83
  "   8D to 12D    59
  "  12D to 16D    32
  "  16D to 20D     8
  "  20D to 24D     8
  "  24D to 28D     7
  "  28D to 32D     0
 above 32D          1

One hundred and fifty-four, considerably more than half, have
inclinations between 3D and 11D, and the mean of the whole number is
about 8D,--slightly greater than the inclination of Mercury, or that of
the plane of the sun's equator. The smallest inclination, that of
Massalia, is 0D 41', and the largest, that of Pallas, is about 35D.
Sixteen minor planets, or six per cent. of the whole number, have
inclinations exceeding 20D. Does any relation obtain between high
inclinations and great eccentricities? These elements in the cases named
above are as follows:

 +------------+--------------+--------------+
 | Asteroid.  | Inclination. | Eccentricity.|
 +------------+--------------+--------------+
 | Pallas     |  34D   42'   |    0.238     |
 | Istria     |  26    30    |    0.353     |
 | Euphrosyne |  26    29    |    0.228     |
 | Anna       |  25    24    |    0.263     |
 | Gallia     |  25    21    |    0.185     |
 | Aethra     |  25     0    |    0.380     |
 | Eukrate    |  24    57    |    0.236     |
 | Eva        |  24    25    |    0.347     |
 | Niobe      |  23    19    |    0.173     |
 | Eunice     |  23    17    |    0.129     |
 | Electra    |  22    55    |    0.208     |
 | Idunna     |  22    31    |    0.164     |
 | Phocea     |  21    35    |    0.255     |
 | Artemis    |  21    31    |    0.175     |
 | Bertha     |  20    59    |    0.085     |
 | Henrietta  |  20    47    |    0.260     |
 +------------+--------------+--------------+

This comparison shows the most inclined orbits to be also very
eccentric; Bertha and Eunice being the only exceptions in the foregoing
list. On the other hand, however, we find over fifty asteroids with
eccentricities exceeding 0.20 whose inclinations are not extraordinary.
The dependence of the phenomena on a common cause can, therefore, hardly
be admitted. At least, the forces which produced the great eccentricity
failed in a majority of cases to cause high inclinations.


8. Longitudes of the Perihelia.

The perihelia of the asteroidal orbits are very unequally distributed;
one hundred and thirty-six--a majority of the whole number
determined--being within the 120D from longitude 290D 50' to 59D 50'.
The maximum occurs between 30D and 60D, where thirty-five perihelia are
found in 30D of longitude.


9. Distribution of the Ascending Nodes.

An inspection of the column containing the longitudes of the ascending
nodes, in Table II., indicates two well-marked maxima, each extending
about sixty degrees, in opposite parts of the heavens.

  I. From 310D to  10D, containing 61 ascending nodes.
 II.   "  120D to 180D,      "     59     "       "
                                  ---
     Making in 120D               120     "       "

A uniform distribution would give 89. An arc of 84D--from 46D to
130D--contains the ascending nodes of all the old planets. This arc, it
will be noticed, is not coincident with either of the maxima found for
the asteroids.


10. The Periods.

Since, according to Kepler's third law, the periods of planets depend
upon their mean distances, the clustering tendency found in the latter
must obtain also in the former. This marked irregularity in the order of
periods is seen below.

 Between 1100 and 1200 days     6 periods.
     "   1200  "  1300   "      7    "
     "   1300  "  1400   "     43    "
     "   1400  "  1500   "     13    "
     "   1500  "  1600   "     46    "
     "   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 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 "Systeme 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.

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 Association 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 passing 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     |   14D 39'   | 101D 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 assigned 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 orbit 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 cluster 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 clusters 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 cluster; and, finally, the
phenomena of luminous meteors indicate the existence of small masses of
matter moving with different velocities in interstellar space.


NOTE B.

THE ORIGIN AND STRUCTURE OF COSMICAL RINGS.

The general theory of cosmical rings and of their arrangement in
sections or clusters 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 assign 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] Massalia was discovered by De Gasparis, at Naples, Sept. 19, 1852,
and independently, the next night, by Chacornac, at Marseilles. The name
was given by the latter.

[3] Astr. Nach., No. 932.

[4] Monthly Notices, vol. xxvii.

[5] Annals of the Obs. of Harv. Coll., 1879.

[6] This ingenious idea may be readily extended. The least distance of
Aethra is less than the present aphelion distance of Mars; and the
maximum aphelion distance of the latter exceeds the perihelion distance
of several known asteroids. Moreover, if we represent the orbits of the
major planets, and also those of the comets of known periods, by
material rings, it is easy to see that the major as well as the minor
planets are all linked together in the manner suggested by D'Arrest.

[7] The effects of Jupiter's disturbing influence will again be resumed.

[8] Not only nebulae are probably unstable, but also many of the sidereal
systems. The Milky Way itself was so regarded by Sir William Herschel.

[9] Menippe, No. 188, is placed in one of the gaps by its calculated
elements; but the fact that it has not been seen since the year of its
discovery, 1878, indicates a probable error in its elements.

[10] The minor planet Andromache, immediately interior to the critical
distance 3.51, has elements somewhat remarkable. With two exceptions,
Aethra (132) and Istria (183), it has the greatest eccentricity
(0.3571),--nearly equal to that of the comet 1867 II. at its last
return. Its perihelion distance is 2.2880, its aphelion 4.7262; hence
the distance from the perihelion to the aphelion of its orbit is greater
than its least distance from the sun, and it crosses the orbits of all
members of the group so far as known; its least distance from the sun
being considerably less than the aphelion of Medusa, and its greatest
exceeding the aphelion of Hilda.

[11] The unit being the sun's distance from the earth.

[12] Annuaire, 1886.



***