Janttabt 15, 1909] 



SCIENCE 



87 



proving the existence of zones of instability 

 surrounding Jupiter's orbit which may ex- 

 tend throughout the plane. The above 

 conditions are approximately satisfied for 

 the small planets (167) Urda, (243) Ida, 

 and (396) whose mean motion is near 

 1 + 3/2 and whose eccentricities and in- 

 clinations are very small; the planet (188) 

 Menippo has a mean motion near 5/2, but 

 an inclination and an eccentricity too large 

 for these considerations to be immediately 

 applicable. Kobb has called the attention 

 of astronomers to the fact that he found 

 the orbit of (153) Hilda to be stable but 

 that the conditions for stability are not 

 satisfied by the motion of (279) Thule; the 

 same writer has shown the motion of the 

 seventh satellite of Jupiter to be stable and 

 that of the eighth imstable, while Moulton 

 has established limits of temporary stability 

 for satellite motion. Levi-Civita's criteria 

 have been studied by Cigala, and those of 

 Lehmann-Filhes for circular motion have 

 been generalized by Frank. Gray has 

 given a resume of the work of Charlier, 

 Hill, Picart, Roche and Schiaparelli on the 

 stability of a swarm of meteorites and of a 

 planet and satellite, and Routh has dis- 

 cussed the motion and stability of a swarm 

 of particles whose center of gravity de- 

 scribes an elliptic orbit of small eccentricity 

 about the sun. Considering a system com- 

 posed of a planet, a rigid ring, and a 

 satellite Bohl has proved that under cer- 

 tain initial conditions the motion can be 

 terminated only by a ring planet collision ; 

 further, that the possibility of the latter 

 collision may be excluded and permanent 

 stability secured. 



The new methods in celestial mechanics 

 have proved their usefulness in computing 

 _the perturbations of those small planets 

 whose period of revolution is approxi- 

 mately commensurable with that of Ju- 

 piter. To enumerate: Simonin has applied 

 Poincare's methods to the case of Hecuba 



and has succeeded in obtaining a very close 

 solution by means of simple expressions; 

 Hill has devoted two memoirs to examples 

 of periodic solution in studies primarily 

 concerned with cases of mean motions re- 

 spectively triple and double (Hecuba type) 

 that of Jupiter; Poineare has shown the 

 essential agreement between his own results 

 and those of Brendel's "Theorie der 

 kleinen Planeten" constructed along the 

 lines of Gylden's method; Hill and An- 

 doyer have applied Delaunay's method to 

 the Hecuba group ; Poineare has exhibited 

 the relations of Simonin 's results to the 

 applications of Gylden's method made by 

 Ludendorff to Hecuba, by Brendel to 

 Hestia, and by Harzer to Hecuba; 

 Sehwarzschild has made a numerical in- 

 vestigation of periodic solutions in the 

 vicinity of the Hecuba orbit ; Wilkens has 

 applied his asymmetric solutions to orbits 

 of the Hecuba type, establishing their 

 stability by Poineare 's method ; and finally 

 Wilkens and De Sitter have studied solu- 

 tions of the Hestia type. A class of 

 periodic solutions was designed by Moulton, 

 and successfully applied to the lunar 

 theory ; independently Gylden and Moulton 

 utilized periodic orbits to explain the 

 Gegenschein; and McCallie, following a 

 suggestion of Hill's, constructed an ex- 

 ample of periodic solutions from the theory 

 of Jupiter and Saturn. Stromgren found 

 that asymptotic motion towards one of the 

 equilateral triangular centers of libration 

 takes place only under exceptional circum- 

 stances, for as a rule the body describes a 

 periodic orbit around this center or recedes 

 indefinitely from it. 



In an exhaustive treatment including 

 all the limitation and libration motions of 

 the special case of the three-body problem 

 when two of the bodies are fixed Charlier 

 has noted two applications : first to the case 

 where a small body passes at great speed 

 through a double star system, and second 



