The Asteroids; Or Minor Planets Between Mars and Jupiter.
Date: 2012-12-06T00:00:00+00:00
Keywords: Asteroids

The Project Gutenberg eBook of The Asteroids; Or Minor Planets Between Mars and Jupiter.

This ebook is for the use of anyone anywhere in the United States and most other parts of the world at no cost and with almost no restrictions whatsoever. You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this ebook or online at www.gutenberg.org. If you are not located in the United States, you will have to check the laws of the country where you are located before using this eBook.

Title: The Asteroids; Or Minor Planets Between Mars and Jupiter.

Author: Daniel Kirkwood

Release date: December 6, 2012 [eBook #41570]
Most recently updated: October 23, 2024

Language: English

Credits: E-text prepared by Paul Clark, sp1nd, and the Online Distributed Proofreading Team (http://www.pgdp.net) from page images generously made available by Internet Archive (http://archive.org)

*** START OF THE PROJECT GUTENBERG EBOOK THE ASTEROIDS; OR MINOR PLANETS BETWEEN MARS AND JUPITER. ***

The Project Gutenberg eBook, The Asteroids, by Daniel Kirkwood

Note:

Images of the original pages are available through Internet Archive. See http://archive.org/details/asteroidsorminor00kirkrich

Transcriber's Note:

Every effort has been made to replicate this text as faithfully as possible, including non-standard spelling and punctuation. Some apparent typographical errors in the indices and names of 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.

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 Astræa; 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 Astræa.

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
Discovery.

Name of
Discoverer.

Place of
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. Astræa | 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. Lætitia | 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. Danaë | 1860, Sept. 9 | Goldschmidt | Paris
62. Erato | 1860, Sept. 14 | Foerster and Lesser | Berlin
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 Safford | Clinton
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. Ægina | 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. Ægle | 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. Æthra | 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. Melibœa | 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. Æmilia | 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. Phædra | 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. Lilæa | 1880, Feb. 16 | Peters | Clinton
214. Aschera | 1880, Feb. 26 | Palisa | Pola
215. Œnone | 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. Cœlestina | 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, π.

4. The longitude of the ascending node, ☊.

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 | π | ☊ | i
149. Medusa | 2.1327 | 1137.7d | 0.1194 | 246°37´ | 342°13´ | 1°6´
244. Sita | 2.1765 | 1172.8 | 0.1370 | 138 | 20837 | 250
228. Agathe | 2.2009 | 1192.6 | 0.2405 | 32923 | 31318 | 233
8. Flora | 2.2014 | 1193.3 | 0.1567 | 3254 | 11018 | 553
43. Ariadne | 2.2033 | 1194.5 | 0.1671 | 27758 | 26435 | 328
254. Augusta | 2.2060 | 1196.8 | 0.1227 | 26047 | 289 | 436
72. Feronia | 2.2661 | 1246.0 | 0.1198 | 30758 | 20749 | 524
40. Harmonia | 2.2673 | 1247.0 | 0.0466 | 054 | 9335 | 416
207. Hedda | 2.2839 | 1260.7 | 0.0301 | 2172 | 2851 | 349
136. Austria | 2.2863 | 1262.7 | 0.0849 | 3166 | 1867 | 933
18. Melpomene | 2.2956 | 1270.4 | 0.2177 | 156 | 1504 | 109
80. Sappho | 2.2962 | 1270.9 | 0.2001 | 35518 | 21844 | 837
261. Prymno | 2.3062 | 1278.4 | 0.0794 | 17935 | 9633 | 338
12. Victoria | 2.3342 | 1302.7 | 0.2189 | 30139 | 23535 | 823
27. Euterpe | 2.3472 | 1313.5 | 0.1739 | 8759 | 9351 | 136
219. Thusnelda | 2.3542 | 1319.4 | 0.2247 | 34034 | 20044 | 1047
163. Erigone | 2.3560 | 1320.9 | 0.1567 | 9346 | 1592 | 442
169. Zelia | 2.3577 | 1322.3 | 0.1313 | 32620 | 35438 | 531
4. Vesta | 2.3616 | 1325.6 | 0.0884 | 25057 | 10329 | 78
186. Celuta | 2.3623 | 1326.2 | 0.1512 | 32724 | 1434 | 136
84. Clio | 2.3629 | 1326.7 | 0.2360 | 33920 | 32728 | 922
51. Nemausa | 2.3652 | 1328.6 | 0.0672 | 17443 | 17552 | 957
220. Stephania | 2.3666 | 1329.8 | 0.2653 | 33253 | 25824 | 735
30. Urania | 2.3667 | 1329.9 | 0.1266 | 3146 | 30812 | 26
105. Artemis | 2.3744 | 1336.4 | 0.1749 | 24238 | 1883 | 2131
113. Amalthea | 2.3761 | 1337.8 | 0.0874 | 19844 | 12311 | 52
115. Thyra | 2.3791 | 1340.3 | 0.1939 | 432 | 3095 | 1135
161. Athor | 2.3792 | 1340.5 | 0.1389 | 31040 | 1827 | 93
172. Baucis | 2.3794 | 1340.6 | 0.1139 | 32923 | 33150 | 102
249. Ilse | 2.3795 | 1340.6 | 0.2195 | 1417 | 33449 | 940
230. Athamantis | 2.3842 | 1344.6 | 0.0615 | 1731 | 23933 | 926
7. Iris | 2.3862 | 1346.4 | 0.2308 | 4123 | 25948 | 528
9. Metis | 2.3866 | 1346.7 | 0.1233 | 714 | 6832 | 536
234. Barbara | 2.3873 | 1347.3 | 0.2440 | 33326 | 1449 | 1522
60. Echo | 2.3934 | 1352.4 | 0.1838 | 9836 | 1925 | 335
63. Ausonia | 2.3979 | 1356.3 | 0.1239 | 27025 | 33758 | 548
25. Phocea | 2.4005 | 1358.5 | 0.2553 | 30248 | 20827 | 2135
192. Nausicaa | 2.4014 | 1359.3 | 0.2413 | 34319 | 16046 | 650
20. Massalia | 2.4024 | 1365.8 | 0.1429 | 997 | 20636 | 041
265. Anna | 2.4096 | 1366.2 | 0.2628 | 22618 | 33526 | 2524
182. Elsa | 2.4157 | 1371.4 | 0.1852 | 5152 | 10630 | 20
142. Polana | 2.4194 | 1374.5 | 0.1322 | 21954 | 31734 | 214
67. Asia | 2.4204 | 1375.4 | 0.1866 | 30635 | 20247 | 559
44. Nysa | 2.4223 | 1377.0 | 0.1507 | 11157 | 13111 | 342
6. Hebe | 2.4254 | 1379.3 | 0.2034 | 1516 | 13843 | 1047
83. Beatrix | 2.4301 | 1383.6 | 0.0859 | 19146 | 2732 | 50
135. Hertha | 2.4303 | 1383.8 | 0.2037 | 32011 | 3443 | 219
131. Vala | 2.4318 | 1385.1 | 0.0683 | 22250 | 6515 | 458
112. Iphigenia | 2.4335 | 1386.6 | 0.1282 | 3389 | 3243 | 237
21. Lutetia | 2.4354 | 1388.2 | 0.1621 | 3274 | 8028 | 35
118. Peitho | 2.4384 | 1390.8 | 0.1608 | 7736 | 4730 | 748
126. Velleda | 2.4399 | 1392.1 | 0.1061 | 34746 | 237 | 256
42. Isis | 2.4401 | 1392.2 | 0.2256 | 31758 | 8428 | 835
19. Fortuna | 2.4415 | 1394.4 | 0.1594 | 313 | 21127 | 133
79. Eurynome | 2.4436 | 1395.2 | 0.1945 | 4422 | 20644 | 437
138. Tolosa | 2.4492 | 1400.0 | 0.1623 | 31139 | 5452 | 314
189. Phthia | 2.4505 | 1401.1 | 0.0356 | 650 | 20322 | 510
11. Parthenope | 2.4529 | 1403.2 | 0.0994 | 3182 | 12511 | 437
178. Belisana | 2.4583 | 1407.8 | 0.1266 | 2780 | 5017 | 25
198. Ampella | 2.4595 | 1408.9 | 0.2266 | 35446 | 26845 | 920
248. Lameia | 2.4714 | 1419.1 | 0.0656 | 24840 | 24634 | 41
17. Thetis | 2.4726 | 1420.1 | 0.1293 | 26137 | 12524 | 536
46. Hestia | 2.5265 | 1466.8 | 0.1642 | 35414 | 18131 | 217
89. Julia | 2.5510 | 1488.2 | 0.1805 | 35313 | 31142 | 1611
232. Russia | 2.5522 | 1489.3 | 0.1754 | 20025 | 15230 | 64
29. Amphitrite | 2.5545 | 1491.3 | 0.0742 | 5623 | 35641 | 67
170. Maria | 2.5549 | 1491.7 | 0.0639 | 9547 | 30120 | 1423
262. Valda | 2.5635 | 1496.4 | 0.2172 | 6142 | 3840 | 746
258. Tyche | 2.5643 | 1499.8 | 0.1966 | 1542 | 2084 | 1450
134. Sophrosyne | 2.5647 | 1500.3 | 0.1165 | 6733 | 34622 | 1136
264. Libussa | 2.5672 | 1502.4 | 0.0925 | 07 | 5023 | 1029
193. Ambrosia | 2.5758 | 1510.0 | 0.2854 | 7052 | 35115 | 1139
13. Egeria | 2.5765 | 1510.6 | 0.0871 | 12010 | 4312 | 1632
5. Astræa | 2.5786 | 1512.4 | 0.1863 | 13457 | 14128 | 519
119. Althea | 2.5824 | 1515.7 | 0.0815 | 1129 | 20357 | 545
157. Dejanira | 2.5828 | 1516.1 | 0.2105 | 10724 | 6231 | 122
101. Helena | 2.5849 | 1518.0 | 0.1386 | 32715 | 34346 | 1011
32. Pomona | 2.5873 | 1520.1 | 0.0830 | 19322 | 22043 | 529
91. Ægina | 2.5895 | 1522.1 | 0.1087 | 8022 | 117 | 28
14. Irene | 2.5896 | 1522.1 | 0.1627 | 18019 | 8648 | 98
111. Ate | 2.5927 | 1524.8 | 0.1053 | 10842 | 30613 | 457
151. Abundantia | 2.5932 | 1525.3 | 0.0356 | 17355 | 3848 | 630
56. Melete | 2.6010 | 1532.2 | 0.2340 | 29450 | 1941 | 82
132. Æthra | 2.6025 | 1533.5 | 0.3799 | 15224 | 2602 | 250
214. Aschera | 2.6111 | 1541.1 | 0.0316 | 11555 | 34230 | 327
70. Panopea | 2.6139 | 1543.6 | 0.1826 | 29949 | 4818 | 1138
194. Procne | 2.6159 | 1545.4 | 0.2383 | 31933 | 15919 | 1824
53. Calypso | 2.6175 | 1546.8 | 0.2060 | 9252 | 14358 | 57
78. Diana | 2.6194 | 1548.5 | 0.2088 | 12142 | 33358 | 840
124. Alceste | 2.6297 | 1557.6 | 0.0784 | 24542 | 18826 | 256
23. Thalia | 2.6306 | 1558.4 | 0.2299 | 12358 | 6745 | 1014
164. Eva | 2.6314 | 1559.1 | 0.3471 | 35932 | 7728 | 2425
15. Eunomia | 2.6437 | 1570.0 | 0.1872 | 2752 | 18826 | 256
37. Fides | 2.6440 | 1570.3 | 0.1758 | 6626 | 821 | 37
66. Maia | 2.6454 | 1571.6 | 0.1750 | 488 | 817 | 36
224. Oceana | 2.6465 | 1572.6 | 0.0455 | 27051 | 35318 | 552
253. Mathilde | 2.6469 | 1572.9 | 0.2620 | 33339 | 1803 | 637
50. Virginia | 2.6520 | 1577.4 | 0.2852 | 109 | 17345 | 248
144. Vibilia | 2.6530 | 1578.4 | 0.2348 | 79 | 7647 | 448
85. Io | 2.6539 | 1579.2 | 0.1911 | 32235 | 20356 | 1153
26. Proserpine | 2.6561 | 1581.1 | 0.0873 | 23625 | 4555 | 336
233. Asterope | 2.6596 | 1584.3 | 0.1010 | 34436 | 22225 | 739
102. Miriam | 2.6619 | 1586.3 | 0.3035 | 35439 | 21158 | 54
240. Vanadis | 2.6638 | 1588.0 | 0.2056 | 5153 | 11454 | 26
73. Clytie | 2.6652 | 1589.3 | 0.0419 | 5755 | 751 | 224
218. Bianca | 2.6653 | 1589.3 | 0.1155 | 23014 | 17050 | 1513
141. Lumen | 2.6666 | 1590.5 | 0.2115 | 1343 | 3197 | 1157
77. Frigga | 2.6680 | 1591.8 | 0.1318 | 5847 | 20 | 228
3. Juno | 2.6683 | 1592.0 | 0.2579 | 5450 | 17053 | 131
97. Clotho | 2.6708 | 1594.3 | 0.2550 | 6532 | 16037 | 1146
75. Eurydice | 2.6720 | 1595.3 | 0.3060 | 33533 | 35956 | 51
145. Adeona | 2.6724 | 1595.4 | 0.1406 | 11753 | 7741 | 1238
204. Callisto | 2.6732 | 1596.4 | 0.1752 | 25745 | 20540 | 819
114. Cassandra | 2.6758 | 1598.8 | 0.1401 | 1536 | 16424 | 455
201. Penelope | 2.6764 | 1599.3 | 0.1818 | 33421 | 1575 | 544
64. Angelina | 2.6816 | 1603.9 | 0.1271 | 12536 | 3114 | 119
98. Ianthe | 2.6847 | 1606.7 | 0.1920 | 14852 | 3547 | 1532
34. Circe | 2.6864 | 1608.3 | 0.1073 | 14841 | 18446 | 527
123. Brunhilda | 2.6918 | 1613.2 | 0.1150 | 7257 | 30828 | 627
166. Rhodope | 2.6927 | 1613.9 | 0.2140 | 3051 | 12933 | 122
109. Felicitas | 2.6950 | 1616.0 | 0.3002 | 561 | 456 | 83
246. Asporina | 2.6994 | 1619.9 | 0.1065 | 25554 | 16235 | 1539
58. Concordia | 2.7004 | 1620.8 | 0.0426 | 18910 | 16120 | 52
103. Hera | 2.7014 | 1621.8 | 0.0803 | 3213 | 13618 | 524
54. Alexandra | 2.7095 | 1629.1 | 0.2000 | 29539 | 31345 | 1147
226. Weringia | 2.7118 | 1631.2 | 0.2048 | 28446 | 13518 | 1550
59. Olympia | 2.7124 | 1631.7 | 0.1189 | 1733 | 17026 | 837
146. Lucina | 2.7189 | 1637.5 | 0.0655 | 22734 | 8416 | 136
45. Eugenia | 2.7205 | 1639.0 | 0.0811 | 2325 | 14757 | 635
210. Isabella | 2.7235 | 1641.7 | 0.1220 | 4422 | 3258 | 518
187. Lamberta | 2.7272 | 1645.0 | 0.2391 | 2144 | 2213 | 1043
180. Garumna | 2.7286 | 1646.3 | 0.1722 | 12556 | 31442 | 054
160. Una | 2.7287 | 1646.4 | 0.0624 | 5557 | 922 | 351
140. Siwa | 2.7316 | 1649.0 | 0.2160 | 30033 | 1072 | 312
110. Lydia | 2.7327 | 1650.0 | 0.0770 | 33649 | 5710 | 60
185. Eunice | 2.7372 | 1654.1 | 0.1292 | 1632 | 15350 | 2317
203. Pompeia | 2.7376 | 1654.5 | 0.0588 | 4251 | 34837 | 313
200. Dynamene | 2.7378 | 1654.6 | 0.1335 | 4638 | 32526 | 656
197. Arete | 2.7390 | 1655.8 | 0.1621 | 32451 | 826 | 848
206. Hersilia | 2.7399 | 1656.5 | 0.0389 | 9544 | 14516 | 346
255. Oppavia | 2.7402 | 1656.6 | 0.0728 | 16915 | 146 | 933
247. Eukrate | 2.7412 | 1657.7 | 0.2387 | 5344 | 020 | 257
38. Leda | 2.7432 | 1659.6 | 0.1531 | 10120 | 29627 | 657
125. Liberatrix | 2.7437 | 1660.0 | 0.0798 | 27329 | 16935 | 438
173. Ino | 2.7446 | 1660.8 | 0.2047 | 1328 | 14834 | 1415
36. Atalanta | 2.7452 | 1661.3 | 0.3023 | 4244 | 35914 | 1842
128. Nemesis | 2.7514 | 1666.9 | 0.1257 | 1634 | 7631 | 616
93. Minerva | 2.7537 | 1669.0 | 0.1405 | 27444 | 54 | 837
127. Johanna | 2.7550 | 1670.3 | 0.0659 | 12237 | 3146 | 817
71. Niobe | 2.7558 | 1671.0 | 0.1732 | 22117 | 31630 | 2319
213. Lilæa | 2.7563 | 1671.4 | 0.1437 | 2814 | 12217 | 647
55. Pandora | 2.7604 | 1675.1 | 0.1429 | 1036 | 1056 | 714
237. Cœlestina | 2.7607 | 1675.5 | 0.0738 | 28249 | 8433 | 946
143. Adria | 2.7619 | 1676.6 | 0.0729 | 22227 | 33342 | 1130
82. Alcmene | 2.7620 | 1676.6 | 0.2228 | 13145 | 2657 | 251
116. Sirona | 2.7669 | 1681.1 | 0.1433 | 15247 | 6426 | 335
1. Ceres | 2.7673 | 1681.4 | 0.0763 | 14938 | 8047 | 1037
88. Thisbe | 2.7673 | 1681.5 | 0.1632 | 30834 | 27754 | 1611
215. Œnone | 2.7679 | 1682.0 | 0.0390 | 34624 | 2525 | 144
2. Pallas | 2.7680 | 1682.1 | 0.2408 | 12212 | 17245 | 3444
39. Lætitia | 2.7680 | 1682.1 | 0.1142 | 38 | 15715 | 1022
41. Daphne | 2.7688 | 1682.8 | 0.2674 | 22033 | 1798 | 1558
177. Irma | 2.7695 | 1683.5 | 0.2370 | 226 | 34917 | 127
148. Gallia | 2.7710 | 1684.8 | 0.1855 | 367 | 14513 | 2521
267. Tirza | 2.7742 | 1687.6 | 0.0986 | 2645 | 7359 | 62
74. Galatea | 2.7770 | 1690.3 | 0.2392 | 818 | 19751 | 40
205. Martha | 2.7771 | 1690.4 | 0.1752 | 2154 | 21212 | 1040
139. Juewa | 2.7793 | 1692.4 | 0.1773 | 16434 | 221 | 1057
28. Bellona | 2.7797 | 1692.7 | 0.1491 | 1241 | 14437 | 922
68. Leto | 2.7805 | 1693.5 | 0.1883 | 34514 | 451 | 758
216. Cleopatra | 2.7964 | 1708.0 | 0.2492 | 32815 | 21549 | 132
99. Dike | 2.7966 | 1708.3 | 0.2384 | 24036 | 4144 | 1353
236. Honoria | 2.7993 | 1710.7 | 0.1893 | 35659 | 18627 | 737
183. Istria | 2.8024 | 1713.4 | 0.3530 | 450 | 14246 | 2633
266. Aline | 2.8078 | 1718.5 | 0.1573 | 2352 | 23618 | 1320
188. Menippe | 2.8211 | 1730.7 | 0.2173 | 30938 | 24144 | 1121
167. Urda | 2.8533 | 1760.4 | 0.0340 | 2964 | 16628 | 211
81. Terpsichore | 2.8580 | 1764.8 | 0.2080 | 491 | 225 | 755
174. Phædra | 2.8600 | 1766.6 | 0.1492 | 25312 | 32849 | 129
243. Ida | 2.8610 | 1767.5 | 0.0419 | 7122 | 32621 | 110
242. Kriemhild | 2.8623 | 1768.7 | 0.1219 | 1231 | 20757 | 1117
129. Antigone | 2.8678 | 1773.9 | 0.2126 | 2424 | 13737 | 1210
217. Eudora | 2.8690 | 1774.9 | 0.3068 | 31441 | 16410 | 1019
158. Coronis | 2.8714 | 1777.2 | 0.0545 | 5656 | 28130 | 10
33. Polyhymnia | 2.8751 | 1780.7 | 0.3349 | 34259 | 919 | 156
195. Euryclea | 2.8790 | 1784.2 | 0.0471 | 11548 | 757 | 71
235. Caroline | 2.8795 | 1784.7 | 0.0595 | 26829 | 6635 | 94
47. Aglaia | 2.8819 | 1786.9 | 0.1317 | 31240 | 4020 | 51
208. Lachrymosa | 2.8926 | 1796.9 | 0.0149 | 12752 | 543 | 148
191. Kolga | 2.8967 | 1800.8 | 0.0876 | 2321 | 15947 | 1129
22. Calliope | 2.9090 | 1801.0 | 0.0193 | 6243 | 447 | 145
155. Scylla | 2.9127 | 1815.7 | 0.2559 | 821 | 4252 | 144
238. Hypatia | 2.9163 | 1819.0 | 0.0946 | 3218 | 18426 | 1228
231. Vindobona | 2.9192 | 1821.7 | 0.1537 | 25323 | 35249 | 510
16. Psyche | 2.9210 | 1823.4 | 0.1392 | 159 | 15036 | 34
179. Clytemnestra | 2.9711 | 1870.6 | 0.1133 | 35539 | 25313 | 747
239. Adrastea | 2.9736 | 1873.0 | 0.2279 | 261 | 18134 | 64
69. Hesperia | 2.9779 | 1877.0 | 0.1712 | 10819 | 18712 | 828
150. Nuwa | 2.9785 | 1877.5 | 0.1307 | 35527 | 20735 | 29
61. Danaë | 2.9855 | 1884.2 | 0.1615 | 3444 | 33411 | 1814
117. Lomia | 2.9907 | 1889.1 | 0.0229 | 4846 | 34939 | 1458
35. Leucothea | 2.9923 | 1890.6 | 0.2237 | 20225 | 35549 | 812
263. Dresda | 3.0120 | 1909.3 | 0.3051 | 30849 | 21756 | 127
221. Eos | 3.0134 | 1910.7 | 0.1028 | 33058 | 14235 | 1051
162. Laurentia | 3.0241 | 1920.8 | 0.1726 | 14552 | 3815 | 64
156. Xantippe | 3.0375 | 1933.7 | 0.2637 | 15558 | 24611 | 729
241. Germania | 3.0381 | 1934.0 | 0.1013 | 3407 | 27228 | 530
256. Walpurga | 3.0450 | 1940.8 | 0.1180 | 24017 | 18335 | 1244
211. Isolda | 3.0464 | 1942.2 | 0.1541 | 7412 | 26529 | 351
96. Ægle | 3.0497 | 1945.3 | 0.1405 | 16310 | 32250 | 167
257. Silesia | 3.0572 | 1952.5 | 0.2555 | 5416 | 3431 | 441
133. Cyrene | 3.0578 | 1953.0 | 0.1398 | 24713 | 3218 | 714
95. Arethusa | 3.0712 | 1965.9 | 0.1447 | 3258 | 24417 | 1254
202. Chryseis | 3.0777 | 1972.1 | 0.0959 | 12946 | 13747 | 848
268. —— | 3.0852 | 1973.9 | 0.1285 | 18448 | 12153 | 225
100. Hecate | 3.0904 | 1984.3 | 0.1639 | 3083 | 12812 | 623
49. Pales | 3.0908 | 1984.7 | 0.2330 | 3115 | 29040 | 38
223. Rosa | 3.0940 | 1987.9 | 0.1186 | 10248 | 490 | 159
52. Europa | 3.0955 | 1988.0 | 0.1098 | 10657 | 12940 | 727
245. Vera | 3.0985 | 1992.1 | 0.1950 | 2529 | 6237 | 510
86. Semele | 3.1015 | 1995.1 | 0.2193 | 2910 | 8745 | 447
159. Æmilia | 3.1089 | 2002.2 | 0.1034 | 10122 | 1359 | 64
48. Doris | 3.1127 | 2005.9 | 0.0649 | 7033 | 18455 | 631
196. Philomela | 3.1137 | 2006.8 | 0.0118 | 30919 | 7324 | 716
130. Electra | 3.1145 | 2007.7 | 0.2132 | 2034 | 1466 | 2257
212. Medea | 3.1157 | 2008.8 | 0.1013 | 5618 | 31516 | 416
120. Lachesis | 3.1211 | 2014.0 | 0.0475 | 2140 | 34251 | 71
181. Eucharis | 3.1226 | 2015.4 | 0.2205 | 9525 | 14445 | 1838
62. Erato | 3.1241 | 2016.9 | 0.1756 | 390 | 12546 | 212
222. Lucia | 3.1263 | 2019.0 | 0.1453 | 2582 | 8011 | 211
137. Melibœa | 3.1264 | 2019.1 | 0.2074 | 30758 | 20422 | 1322
165. Loreley | 3.1269 | 2019.6 | 0.0734 | 22350 | 3046 | 1012
251. Sophia | 3.1315 | 2024.1 | 0.1243 | 777 | 1576 | 1020
24. Themis | 3.1357 | 2028.1 | 0.1242 | 1448 | 3549 | 049
152. Atala | 3.1362 | 2028.6 | 0.0862 | 8423 | 4129 | 1212
10. Hygeia | 3.1366 | 2029.1 | 0.1156 | 2372 | 28538 | 349
259. Aletheia | 3.1369 | 2029.3 | 0.1176 | 24145 | 8832 | 1040
227. Philosophia | 3.1393 | 2031.6 | 0.2131 | 22623 | 33052 | 916
147. Protogenea | 3.1393 | 2031.6 | 0.0247 | 2538 | 25116 | 154
171. Ophelia | 3.1432 | 2035.4 | 0.1168 | 14359 | 10110 | 234
209. Dido | 3.1436 | 2035.9 | 0.0637 | 25733 | 20 | 715
31. Euphrosyne | 3.1468 | 2039.0 | 0.2228 | 9326 | 3131 | 2627
90. Antiope | 3.1475 | 2039.7 | 0.1645 | 30115 | 7129 | 217
104. Clymene | 3.1507 | 2042.7 | 0.1579 | 5932 | 4332 | 254
57. Mnemosyne | 3.1510 | 2043.0 | 0.1145 | 5325 | 2002 | 1512
250. Bettina | 3.1524 | 2044.3 | 0.1302 | 8728 | 2612 | 1254
252. Clementina | 3.1552 | 2047.1 | 0.0837 | 3558 | 20819 | 102
94. Aurora | 3.1602 | 2052.0 | 0.0827 | 4846 | 49 | 84
106. Dione | 3.1670 | 2058.6 | 0.1788 | 2557 | 6314 | 438
199. Byblis | 3.1777 | 2069.0 | 0.1687 | 26120 | 8952 | 1522
92. Undina | 3.1851 | 2076.3 | 0.1024 | 33127 | 10252 | 957
184. Deiopea | 3.1883 | 2079.4 | 0.0725 | 16922 | 33618 | 112
176. Idunna | 3.1906 | 2081.6 | 0.1641 | 2034 | 20113 | 2231
154. Bertha | 3.1976 | 2088.5 | 0.0788 | 19047 | 3735 | 2059
108. Hecuba | 3.2113 | 2101.0 | 0.1005 | 17349 | 35217 | 424
122. Gerda | 3.2177 | 2108.2 | 0.0415 | 20345 | 17843 | 136
168. Sibylla | 3.3765 | 2266.2 | 0.0707 | 1126 | 20947 | 433
225. Henrietta | 3.4007 | 2277.8 | 0.2661 | 29913 | 20045 | 2045
229. Adelinda | 3.4129 | 2302.9 | 0.1562 | 3327 | 3049 | 211
76. Freia | 3.4140 | 2304.1 | 0.1700 | 9049 | 2125 | 23
260. Huberta | 3.4212 | 2311.5 | 0.1113 | 31322 | 16848 | 618
65. Maximiliana | 3.4270 | 2317.2 | 0.1097 | 26036 | 15850 | 329
121. Hermione | 3.4535 | 2344.2 | 0.1255 | 35750 | 7646 | 736
87. Sylvia | 3.4833 | 2374.5 | 0.0922 | 33348 | 7549 | 1055
107. Camilla | 3.4847 | 2376.0 | 0.0756 | 11553 | 17618 | 954
175. Andromache | 3.5071 | 2399.0 | 0.3476 | 2930 | 2335 | 346
190. Ismene | 3.9471 | 2864.3 | 0.1634 | 10539 | 1770 | 67
153. Hilda | 3.9523 | 2869.9 | 0.1721 | 28547 | 22820 | 755

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 Æthra (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

Third

2⁄5, 4⁄7, 5⁄8

⎧
⎨
⎩2.82
3.58
3.80

Fourth

3⁄7, 5⁄9, 7⁄11

⎧
⎨
⎩2.95
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 120°. 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 Æthra 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 180°. 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 Æthra (132). Comparing these with the orbit of the second comet of 1867 we have

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

The orbit of Æthra, 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 | 0° to | 4° | 70
" | 4° to | 8° | 83
" | 8° to | 12° | 59
" | 12° to | 16° | 32
" | 16° to | 20° | 8
" | 20° to | 24° | 8
" | 24° to | 28° | 7
" | 28° to | 32° | 0
above 32° | 1

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

Asteroid. | Inclination. | Eccentricity.
Pallas | 34°42´ | 0.238
Istria | 2630 | 0.353
Euphrosyne | 2629 | 0.228
Anna | 2524 | 0.263
Gallia | 2521 | 0.185
Æthra | 250 | 0.380
Eukrate | 2457 | 0.236
Eva | 2425 | 0.347
Niobe | 2319 | 0.173
Eunice | 2317 | 0.129
Electra | 2255 | 0.208
Idunna | 2231 | 0.164
Phocea | 2135 | 0.255
Artemis | 2131 | 0.175
Bertha | 2059 | 0.085
Henrietta | 2047 | 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 120° from longitude 290° 50´ to 59° 50´. The maximum occurs between 30° and 60°, where thirty-five perihelia are found in 30° 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 | 310° to | 10°, | containing | 61 | ascending | nodes.
II. | " | 120° to | 180°, | " | 59 | " | "
| Making in 120° | 120 | " | "

A uniform distribution would give 89. An arc of 84°—from 46° to 130°—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 "Système du Monde." The author, in his celebrated seventh note in the second volume of that work, explained the origin of these bodies by assuming that the primitive ring from which they were formed, instead of collecting into a single sphere, as in the case of the major planets, broke up into four distinct masses. But the form and extent of the cluster as now known, as well as the observed facts bearing on the constitution of Saturn's ring, seem to require a modification of Laplace's theory. Throughout the greater part of the interval between Mars and Jupiter an almost continuous succession of small planetary masses—not nebulous rings—appears to have been abandoned at the solar equator. The entire cluster, distributed throughout a space whose outer radius exceeds the inner by more than two hundred millions of miles, could not have originated, as supposed by Laplace, in a single nebulous zone the different parts of which revolved with the same angular velocity. The following considerations may furnish a suggestion in regard to the mode in which these bodies were separated from the equator of the solar nebula.

(a) The perihelion distance of Jupiter is 4.950, while the aphelion distance of Hilda is 4.623. If, therefore, the sun once extended to the latter, the central attraction of its mass on an equatorial particle was but five times greater than Jupiter's perihelion influence on the same. It is easy to see, then, that this "giant planet" would produce enormous tidal elevations in the solar mass.

(b) The centrifugal force would be greatest at the crest of this tidal wave.

(c) Three periods of solar revolution were then about equal to two periods of Jupiter. The disturbing influence of the planet would therefore be increased at each conjunction with this protuberance. The ultimate separation (not of a ring but) of a planetary mass would be the probable result of these combined and accumulating forces.

12. Variability of Certain Asteroids.

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 | 14°39´ | 101°48´
Longitude of ascending node | 11356 | 13327
Inclination | 10 | 66
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-Filhès, 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 Æthra, 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 Æthra 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 nebulæ 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, Æthra (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.

*** END OF THE PROJECT GUTENBERG EBOOK THE ASTEROIDS; OR MINOR PLANETS BETWEEN MARS AND JUPITER. ***

Updated editions will replace the previous one—the old editions will be renamed.

Creating the works from print editions not protected by U.S. copyright law means that no one owns a United States copyright in these works, so the Foundation (and you!) can copy and distribute it in the United States without permission and without paying copyright royalties. Special rules, set forth in the General Terms of Use part of this license, apply to copying and distributing Project Gutenberg™ electronic works to protect the PROJECT GUTENBERG™ concept and trademark. Project Gutenberg is a registered trademark, and may not be used if you charge for an eBook, except by following the terms of the trademark license, including paying royalties for use of the Project Gutenberg trademark. If you do not charge anything for copies of this eBook, complying with the trademark license is very easy. You may use this eBook for nearly any purpose such as creation of derivative works, reports, performances and research. Project Gutenberg eBooks may be modified and printed and given away—you may do practically ANYTHING in the United States with eBooks not protected by U.S. copyright law. Redistribution is subject to the trademark license, especially commercial redistribution.

START: FULL LICENSE

THE FULL PROJECT GUTENBERG LICENSE

PLEASE READ THIS BEFORE YOU DISTRIBUTE OR USE THIS WORK

To protect the Project Gutenberg™ mission of promoting the free distribution of electronic works, by using or distributing this work (or any other work associated in any way with the phrase “Project Gutenberg”), you agree to comply with all the terms of the Full Project Gutenberg™ License available with this file or online at www.gutenberg.org/license.

Section 1. General Terms of Use and Redistributing Project Gutenberg™ electronic works

1.A. By reading or using any part of this Project Gutenberg™ electronic work, you indicate that you have read, understand, agree to and accept all the terms of this license and intellectual property (trademark/copyright) agreement. If you do not agree to abide by all the terms of this agreement, you must cease using and return or destroy all copies of Project Gutenberg™ electronic works in your possession. If you paid a fee for obtaining a copy of or access to a Project Gutenberg™ electronic work and you do not agree to be bound by the terms of this agreement, you may obtain a refund from the person or entity to whom you paid the fee as set forth in paragraph 1.E.8.

1.B. “Project Gutenberg” is a registered trademark. It may only be used on or associated in any way with an electronic work by people who agree to be bound by the terms of this agreement. There are a few things that you can do with most Project Gutenberg™ electronic works even without complying with the full terms of this agreement. See paragraph 1.C below. There are a lot of things you can do with Project Gutenberg™ electronic works if you follow the terms of this agreement and help preserve free future access to Project Gutenberg™ electronic works. See paragraph 1.E below.

1.C. The Project Gutenberg Literary Archive Foundation (“the Foundation” or PGLAF), owns a compilation copyright in the collection of Project Gutenberg™ electronic works. Nearly all the individual works in the collection are in the public domain in the United States. If an individual work is unprotected by copyright law in the United States and you are located in the United States, we do not claim a right to prevent you from copying, distributing, performing, displaying or creating derivative works based on the work as long as all references to Project Gutenberg are removed. Of course, we hope that you will support the Project Gutenberg™ mission of promoting free access to electronic works by freely sharing Project Gutenberg™ works in compliance with the terms of this agreement for keeping the Project Gutenberg™ name associated with the work. You can easily comply with the terms of this agreement by keeping this work in the same format with its attached full Project Gutenberg™ License when you share it without charge with others.

1.D. The copyright laws of the place where you are located also govern what you can do with this work. Copyright laws in most countries are in a constant state of change. If you are outside the United States, check the laws of your country in addition to the terms of this agreement before downloading, copying, displaying, performing, distributing or creating derivative works based on this work or any other Project Gutenberg™ work. The Foundation makes no representations concerning the copyright status of any work in any country other than the United States.

1.E. Unless you have removed all references to Project Gutenberg:

1.E.1. The following sentence, with active links to, or other immediate access to, the full Project Gutenberg™ License must appear prominently whenever any copy of a Project Gutenberg™ work (any work on which the phrase “Project Gutenberg” appears, or with which the phrase “Project Gutenberg” is associated) is accessed, displayed, performed, viewed, copied or distributed:

This eBook is for the use of anyone anywhere in the United States and most other parts of the world at no cost and with almost no restrictions whatsoever. You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this eBook or online at www.gutenberg.org. If you are not located in the United States, you will have to check the laws of the country where you are located before using this eBook.

1.E.2. If an individual Project Gutenberg™ electronic work is derived from texts not protected by U.S. copyright law (does not contain a notice indicating that it is posted with permission of the copyright holder), the work can be copied and distributed to anyone in the United States without paying any fees or charges. If you are redistributing or providing access to a work with the phrase “Project Gutenberg” associated with or appearing on the work, you must comply either with the requirements of paragraphs 1.E.1 through 1.E.7 or obtain permission for the use of the work and the Project Gutenberg™ trademark as set forth in paragraphs 1.E.8 or 1.E.9.

1.E.3. If an individual Project Gutenberg™ electronic work is posted with the permission of the copyright holder, your use and distribution must comply with both paragraphs 1.E.1 through 1.E.7 and any additional terms imposed by the copyright holder. Additional terms will be linked to the Project Gutenberg™ License for all works posted with the permission of the copyright holder found at the beginning of this work.

1.E.4. Do not unlink or detach or remove the full Project Gutenberg™ License terms from this work, or any files containing a part of this work or any other work associated with Project Gutenberg™.

1.E.5. Do not copy, display, perform, distribute or redistribute this electronic work, or any part of this electronic work, without prominently displaying the sentence set forth in paragraph 1.E.1 with active links or immediate access to the full terms of the Project Gutenberg™ License.

1.E.6. You may convert to and distribute this work in any binary, compressed, marked up, nonproprietary or proprietary form, including any word processing or hypertext form. However, if you provide access to or distribute copies of a Project Gutenberg™ work in a format other than “Plain Vanilla ASCII” or other format used in the official version posted on the official Project Gutenberg™ website (www.gutenberg.org), you must, at no additional cost, fee or expense to the user, provide a copy, a means of exporting a copy, or a means of obtaining a copy upon request, of the work in its original “Plain Vanilla ASCII” or other form. Any alternate format must include the full Project Gutenberg™ License as specified in paragraph 1.E.1.

1.E.7. Do not charge a fee for access to, viewing, displaying, performing, copying or distributing any Project Gutenberg™ works unless you comply with paragraph 1.E.8 or 1.E.9.

1.E.8. You may charge a reasonable fee for copies of or providing access to or distributing Project Gutenberg™ electronic works provided that:

- • You pay a royalty fee of 20% of the gross profits you derive from the use of Project Gutenberg™ works calculated using the method you already use to calculate your applicable taxes. The fee is owed to the owner of the Project Gutenberg™ trademark, but he has agreed to donate royalties under this paragraph to the Project Gutenberg Literary Archive Foundation. Royalty payments must be paid within 60 days following each date on which you prepare (or are legally required to prepare) your periodic tax returns. Royalty payments should be clearly marked as such and sent to the Project Gutenberg Literary Archive Foundation at the address specified in Section 4, “Information about donations to the Project Gutenberg Literary Archive Foundation.”

- • You provide a full refund of any money paid by a user who notifies you in writing (or by e-mail) within 30 days of receipt that s/he does not agree to the terms of the full Project Gutenberg™ License. You must require such a user to return or destroy all copies of the works possessed in a physical medium and discontinue all use of and all access to other copies of Project Gutenberg™ works.

- • You provide, in accordance with paragraph 1.F.3, a full refund of any money paid for a work or a replacement copy, if a defect in the electronic work is discovered and reported to you within 90 days of receipt of the work.

- • You comply with all other terms of this agreement for free distribution of Project Gutenberg™ works.

1.E.9. If you wish to charge a fee or distribute a Project Gutenberg™ electronic work or group of works on different terms than are set forth in this agreement, you must obtain permission in writing from the Project Gutenberg Literary Archive Foundation, the manager of the Project Gutenberg™ trademark. Contact the Foundation as set forth in Section 3 below.

1.F.

1.F.1. Project Gutenberg volunteers and employees expend considerable effort to identify, do copyright research on, transcribe and proofread works not protected by U.S. copyright law in creating the Project Gutenberg™ collection. Despite these efforts, Project Gutenberg™ electronic works, and the medium on which they may be stored, may contain “Defects,” such as, but not limited to, incomplete, inaccurate or corrupt data, transcription errors, a copyright or other intellectual property infringement, a defective or damaged disk or other medium, a computer virus, or computer codes that damage or cannot be read by your equipment.

1.F.2. LIMITED WARRANTY, DISCLAIMER OF DAMAGES - Except for the “Right of Replacement or Refund” described in paragraph 1.F.3, the Project Gutenberg Literary Archive Foundation, the owner of the Project Gutenberg™ trademark, and any other party distributing a Project Gutenberg™ electronic work under this agreement, disclaim all liability to you for damages, costs and expenses, including legal fees. YOU AGREE THAT YOU HAVE NO REMEDIES FOR NEGLIGENCE, STRICT LIABILITY, BREACH OF WARRANTY OR BREACH OF CONTRACT EXCEPT THOSE PROVIDED IN PARAGRAPH 1.F.3. YOU AGREE THAT THE FOUNDATION, THE TRADEMARK OWNER, AND ANY DISTRIBUTOR UNDER THIS AGREEMENT WILL NOT BE LIABLE TO YOU FOR ACTUAL, DIRECT, INDIRECT, CONSEQUENTIAL, PUNITIVE OR INCIDENTAL DAMAGES EVEN IF YOU GIVE NOTICE OF THE POSSIBILITY OF SUCH DAMAGE.

1.F.3. LIMITED RIGHT OF REPLACEMENT OR REFUND - If you discover a defect in this electronic work within 90 days of receiving it, you can receive a refund of the money (if any) you paid for it by sending a written explanation to the person you received the work from. If you received the work on a physical medium, you must return the medium with your written explanation. The person or entity that provided you with the defective work may elect to provide a replacement copy in lieu of a refund. If you received the work electronically, the person or entity providing it to you may choose to give you a second opportunity to receive the work electronically in lieu of a refund. If the second copy is also defective, you may demand a refund in writing without further opportunities to fix the problem.

1.F.4. Except for the limited right of replacement or refund set forth in paragraph 1.F.3, this work is provided to you ‘AS-IS’, WITH NO OTHER WARRANTIES OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO WARRANTIES OF MERCHANTABILITY OR FITNESS FOR ANY PURPOSE.

1.F.5. Some states do not allow disclaimers of certain implied warranties or the exclusion or limitation of certain types of damages. If any disclaimer or limitation set forth in this agreement violates the law of the state applicable to this agreement, the agreement shall be interpreted to make the maximum disclaimer or limitation permitted by the applicable state law. The invalidity or unenforceability of any provision of this agreement shall not void the remaining provisions.

1.F.6. INDEMNITY - You agree to indemnify and hold the Foundation, the trademark owner, any agent or employee of the Foundation, anyone providing copies of Project Gutenberg™ electronic works in accordance with this agreement, and any volunteers associated with the production, promotion and distribution of Project Gutenberg™ electronic works, harmless from all liability, costs and expenses, including legal fees, that arise directly or indirectly from any of the following which you do or cause to occur: (a) distribution of this or any Project Gutenberg™ work, (b) alteration, modification, or additions or deletions to any Project Gutenberg™ work, and (c) any Defect you cause.

Section 2. Information about the Mission of Project Gutenberg™

Project Gutenberg™ is synonymous with the free distribution of electronic works in formats readable by the widest variety of computers including obsolete, old, middle-aged and new computers. It exists because of the efforts of hundreds of volunteers and donations from people in all walks of life.

Volunteers and financial support to provide volunteers with the assistance they need are critical to reaching Project Gutenberg™’s goals and ensuring that the Project Gutenberg™ collection will remain freely available for generations to come. In 2001, the Project Gutenberg Literary Archive Foundation was created to provide a secure and permanent future for Project Gutenberg™ and future generations. To learn more about the Project Gutenberg Literary Archive Foundation and how your efforts and donations can help, see Sections 3 and 4 and the Foundation information page at www.gutenberg.org.

Section 3. Information about the Project Gutenberg Literary Archive Foundation

The Project Gutenberg Literary Archive Foundation is a non-profit 501(c)(3) educational corporation organized under the laws of the state of Mississippi and granted tax exempt status by the Internal Revenue Service. The Foundation’s EIN or federal tax identification number is 64-6221541. Contributions to the Project Gutenberg Literary Archive Foundation are tax deductible to the full extent permitted by U.S. federal laws and your state’s laws.

The Foundation’s business office is located at 809 North 1500 West, Salt Lake City, UT 84116, (801) 596-1887. Email contact links and up to date contact information can be found at the Foundation’s website and official page at www.gutenberg.org/contact

Section 4. Information about Donations to the Project Gutenberg Literary Archive Foundation

Project Gutenberg™ depends upon and cannot survive without widespread public support and donations to carry out its mission of increasing the number of public domain and licensed works that can be freely distributed in machine-readable form accessible by the widest array of equipment including outdated equipment. Many small donations ($1 to $5,000) are particularly important to maintaining tax exempt status with the IRS.

The Foundation is committed to complying with the laws regulating charities and charitable donations in all 50 states of the United States. Compliance requirements are not uniform and it takes a considerable effort, much paperwork and many fees to meet and keep up with these requirements. We do not solicit donations in locations where we have not received written confirmation of compliance. To SEND DONATIONS or determine the status of compliance for any particular state visit www.gutenberg.org/donate.

While we cannot and do not solicit contributions from states where we have not met the solicitation requirements, we know of no prohibition against accepting unsolicited donations from donors in such states who approach us with offers to donate.

International donations are gratefully accepted, but we cannot make any statements concerning tax treatment of donations received from outside the United States. U.S. laws alone swamp our small staff.

Please check the Project Gutenberg web pages for current donation methods and addresses. Donations are accepted in a number of other ways including checks, online payments and credit card donations. To donate, please visit: www.gutenberg.org/donate.

Section 5. General Information About Project Gutenberg™ electronic works

Professor Michael S. Hart was the originator of the Project Gutenberg™ concept of a library of electronic works that could be freely shared with anyone. For forty years, he produced and distributed Project Gutenberg™ eBooks with only a loose network of volunteer support.

Project Gutenberg™ eBooks are often created from several printed editions, all of which are confirmed as not protected by copyright in the U.S. unless a copyright notice is included. Thus, we do not necessarily keep eBooks in compliance with any particular paper edition.

Most people start at our website which has the main PG search facility: www.gutenberg.org.

This website includes information about Project Gutenberg™, including how to make donations to the Project Gutenberg Literary Archive Foundation, how to help produce our new eBooks, and how to subscribe to our email newsletter to hear about new eBooks.

Table of Contents

- THE ASTEROIDS, OR MINOR PLANETS BETWEEN MARS AND JUPITER.

- PREFACE.

- CONTENTS.

- PART I.

- THE ASTEROIDS, OR MINOR PLANETS BETWEEN MARS AND JUPITER.

- 1. Introductory. PLANETARY DISCOVERIES BEFORE THE ASTEROIDS WERE KNOWN.

- 2. Discovery of the First Asteroids.

- TABLE I. The Asteroids in the Order of their Discovery.

- 3. Remarks on Table I.

- 4. Mode of Discovery.

- NAMES AND SYMBOLS.

- 5. Magnitudes of the Asteroids.

- 6. The Orbits of the Asteroids.

- TABLE II. Elements of the Asteroids.

- PART II.

- DISCUSSION OF THE FACTS IN TABLE II.

- 1. Extent of the Zone.

- 2. The Small Mass of the Asteroids.

- 3. The Limits of Perihelion Distance.

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

- COMMENSURABILITY OF PERIODS.

- 5. Relations between certain Adjacent Orbits.

- 6. The Eccentricities.

- 7. The Inclinations.

- 8. Longitudes of the Perihelia.

- 9. Distribution of the Ascending Nodes.

- 10. The Periods.

- 11. Origin of the Asteroids.

- 12. Variability of Certain Asteroids.

- 13. The Average Asteroid Orbit.

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

- APPENDIX.

- NOTE A. THE POSSIBLE EXISTENCE OF ASTEROIDS IN UNDISCOVERED RINGS.

- NOTE B. THE ORIGIN AND STRUCTURE OF COSMICAL RINGS.

- I.

- II.

- THE END.

- FOOTNOTES:

- THE FULL PROJECT GUTENBERG LICENSE