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SCIENCE 



[N. S. Vol. XXXIII. No. 838 



been said concerning the stability of libra- 

 tions around the ratio 2:1, we might be 

 tempted to conclude that the stability of 

 these new asteroids would be still more 

 doubtful. This is not so. When the mean 

 period is the same as that of Jupiter, the 

 librations are very nearly independent of 

 the eccentricity and inclination unless 

 these are very large; and conversely, the 

 latter are but little dependent on the libra- 

 tions of the mean period. It is the only 

 ease in which this independence occurs. 

 The conditions governing these librations 

 are so different that a separate investiga- 

 tion of them is necessary. I had hoped to 

 lay before you the results which I have so 

 far reached in determining the possible 

 motions of asteroids which circulate with 

 the same mean period as Jupiter. In 

 order to do so it would be necessary to ex- 

 ceed greatly the time allotted to me. I 

 must content myself with the remark that 

 much larger deviations from the triangu- 

 lar positions appear to be possible and that 

 these may become so extended that a single 

 asteroid may go round both triangular 

 points, passing from one to the other on the 

 opposite side of the sun from Jupiter, and 

 yet not at any time approach very closely 

 to the planet. "While describing such an 

 orbit the distance of the asteroid from the 

 sun would vary but little. 



In conclusion, a few words must be said 

 concerning the present condition of the ob- 

 servational material and its future needs. 

 There are about 680 bodies whose orbits are 

 more or less well defined. "We have seen 

 that the distribution is by no means uni- 

 form. "With the number on hand we have 

 perhaps sufficient material to determine 

 the general law which governs the arrange- 

 ment of the orbits. This law is, however, 

 only one of the problems which the system 

 presents. The student of celestial mechan- 

 ics needs more asteroids in special regions. 



Reference has been made to the abundance 

 of known orbits on the inner side of the gap 

 which corresponds to twice the period of 

 Jupiter and the lack of material on the 

 outer side. Every asteroid added to 

 those very near the gap will be wel- 

 come, but there is a real need for more 

 on the outer side. In any case new 

 bodies whose periods are close to this 

 ratio should be observed until the eccentri- 

 city and inclination can be obtained with 

 moderate accuracy. In the course of the 

 search it is possible that one may be found 

 which is within the narrow limits of sta- 

 bility (if the distance between these limits 

 is not zero) of libra tion. At every opposi- 

 tion a few observations of all the known 

 asteroids near the gap should be made, so 

 that the orbits may be verified and the 

 long period terms compared with theory. 

 It will perhaps be advisable to limit the 

 number of asteroids chosen for continuous 

 observation. If so, a selection can be made 

 which will give types of small, moderate 

 and large inclinations and eccentricities. 



A reference to the general chart of dis- 

 tribution of the mean periods will indi- 

 cate what observational material is needed 

 at the other gaps. Aside from the aster- 

 oids whose periods have the features which 

 I have just described, every new discovery 

 of a very large eccentricity of inclination 

 should be retained. I would suggest that 

 eccentricities above three tenths or inclina- 

 tions to the orbit of Jupiter above 20° would 

 perhaps not unduly tax the capacity of 

 those observers who have made and are 

 making valuable additions to our store of 

 knowledge of these bodies. 



Finally every asteroid whose period in- 

 dicates that it is outside the main stream 

 or near its edges should not be lost. More 

 particularly the search might be directed 

 towards the discovery of bodies having the 

 same mean distance as Jupiter, but with 



