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^—i 1 n v =o, 



where i, i', are any integers, 



w a =the mean daily motion of a planetary mass between the orbits of 

 Mars and Jupiter, 



w v =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 sun. 





(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 three cases : 



(j?) i=3,tf = 7 



(ij> i = 5, i' = 9 



(0) i==7, i' = ll 



These particular portions of the zone will be separately considered. 



7. At the distance 3.27 — which falls between the orbits of Bertha 

 and Johanna, — a planetary mass would make precisely two revolutions 

 while Jupiter completes one. It is obvious, therefore, that all its con-' 

 junctions with that planet, for an indefinite period, must occur in the 

 same parts of its path. Consequently its orbit would become more and 

 more eccentric, until, if the nebulous ring had considerable density, the 

 disturbed matter would be brought into contact, either in aphelion or 

 perihelion, with masses having somewhat different velocities. The plan- 



