SCIENCE 



[N. S. Vol. XXIX. No. 733 



to the generalization of those conditions 

 which result in the moon revolving about 

 the sun if the earth and sun. become fixed 

 centers. Charlier has also considered the 

 relatione of the two-body problem to the 

 two-center problem, and has pointed out 

 the advantages of solutions of the latter as 

 intermediate orbits in the asteroid problem. 

 The case where one fixed center attracts 

 and the other repels was worked out in 

 detail by Woller; and Hiltebeitel applied 

 the method of Charlier 's work to the quali- 

 tative discussion of the most general two- 

 center problem admitting of separation of 

 the variables. 



Hill has prepared a number of examples 

 of Gylden's periplegmatic orbits, some of 

 which are periodic. The construction of 

 the solutions calls for elliptic functions, 

 Lindstedt's series, and sequences of De- 

 launay transformations. These examples 

 of Hill have been generalized in several 

 directions, in one of which certain of 

 Painleve's new transcendental functions 

 find application. For the case of two 

 nearly equal bodies and a third infini- 

 tesimal body Pavannini found a new cate- 

 gory of periodic solutions which have been 

 extended to the restricted problem of four 

 bodies. Andoyer's memoir on the relative 

 equilibrium of n bodies has been made by 

 him the basis of a study of periodic solu- 

 tions in the vicinity of positions of relative 

 equilibrium under forces varying as the 

 masses and any power of the distances. 

 Longley has constructed the only orbits 

 (one direct the other retrograde) of pre- 

 assigned period in the plane it-body prob- 

 lem which consist of an infinitesimal body 

 revolving around one oi n — 1 finite masses 

 which are in periodic motion. For the 

 plane m-body problem having the same dis- 

 tribution of masses as the solar system. 

 Griffin found a class of periodic solutions 

 of which he has made numerical applica- 

 tion to the three inner satellites of Jupiter. 



FORMAL AND QUALITATIVE RESOLUTION OP 

 THE PROBLEM 



By generalizing somewhat the theory of 

 periodic and asymptotic solutions by which 

 Poincare established the divergence of 

 Lindstedt's series von Zeipel has been able 

 to study the series, however great the mu- 

 tual inclination of the orbits may be. He 

 derived the following necessary and suffi- 

 cient conditions for the existence of the 

 series: first, that the orbits be nearly cir- 

 cular; second, that a certain biquadratic 

 equation have real and unequal roots. 

 Von Zeipel found that if the inclination of 

 an asteroid exceeds a certain limit (about 

 30°, slightly variable) the series of Lind- 

 stedt cease to exist; and he remarked that 

 it is perhaps permissible to see in this 

 theorem, although Lindstedt's series are 

 only semi-convergent, the cause of the sur- 

 prising fact that among five hundred 

 asteroids there exists but one (Pallas) 

 whose inclination exceeds 30°. 



Hill has extended Delaunay's method to 

 the general problem of planetary motion, 

 and, employing the fundamental concep- 

 tions of Gylden, he has indicated a practic- 

 able way for its application, in two 

 memoirs on integrals of planetary motion, 

 suitable for an indefinite length of time. 

 Charlier has discussed the properties of the 

 general solution in trigonometric series by 

 supposing it to have been derived from the 

 integration of the Hamilton-Jaeobi equa- 

 tion. For constructing solutions in the 

 form of trigonometric series, "VVhittaker has 

 devised a method, closely analogous to De- 

 launay's, and consisting essentially in the 

 repeated application of contact transforma- 

 tions which ultimately reduce the problem 

 to the equilibrium problem. Bohlin has 

 just published the concluding memoirs of 

 a remarkable series of investigations which 

 have culminated in a non-existence theorem 

 quoted in a previous paragraph, and in his 

 new astronomical series for the distances 



