. Annual report of the Board of Regents of the Smithsonian Institution. Smithsonian Institution; Smithsonian Institution. Archives; Discoveries in science. THE ASTEROIDS BETWEEN MARS AND JUPITER, 365 G. Distribution of the asteroids.—As long: since as 1857, when the number of known asteroids was less than 50, it was interred from phys- ical considerations by the author of this paper that great irregularity must obtain in the distribution of these bodies, and that gapsor chasms would be found in those parts of the zone where the periods of asteroids ?would be commensurable with tbat of Jupiter.
. Annual report of the Board of Regents of the Smithsonian Institution. Smithsonian Institution; Smithsonian Institution. Archives; Discoveries in science. THE ASTEROIDS BETWEEN MARS AND JUPITER, 365 G. Distribution of the asteroids.—As long: since as 1857, when the number of known asteroids was less than 50, it was interred from phys- ical considerations by the author of this paper that great irregularity must obtain in the distribution of these bodies, and that gapsor chasms would be found in those parts of the zone where the periods of asteroids ?would be commensurable with tbat of Jupiter. To verify this theory every addition to the group was watched with interest. In 18GG, when the number had increased to SS, and the agreement between theory and ob- servation had become quite marked, the attention of astronomers was called to the coincidence by a paper read at the Buffalo meeting of the American Association for the Advancement of Science. The comparison of fact and hypothesis has been continued to the present time; and it is now proposed to show that recent discoveries have confirmed the theory of an irregular distribution. Assuming the equation ?i n^—i1 nv=o, where i, i', are any integers, wa=the mean daily motion of a planetary mass between the orbits of Mars and Jupiter, wv=the mean daily motion of Jupiter=299".1286, and assigning values at pleasure to i and i', we shall find those portions of the asteroid zone in which the periods would be commensurable with that of Jupiter. The following table contains all such instances when i' — i <5. The consideration of less simple cases of commeusurability may be omitted. Corresponding distances from the (1.) When i' —t = l, t = 1, and i' =2, or 2 periods of an asteroid = 1 of Jupiter (2.) i'—i — 2, we 'nave two cases: (a) i = l, t' = 3 (8)i = 3, i' = 5 (3.) When i' — i = 3, there are three cases: (y) i = 2,i' = 5 (6) i = 4, i' = 7 («) i=5, i'— 8 (-1.) When i' — •£ = 4, we have also
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