90 



SCIENCE 



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



lectures announced tlie opinion that the 

 initial conditions which constrain a col- 

 lision of at least two of the three bodies at 

 the end of a finite time, satisfy two distinct 

 analytical relations, which reduce to one in 

 the case of plane motion. These analytical 

 relations whose existence Painleve divined 

 have been disclosed by the brilliant re- 

 searches of the two Italian mathematicians, 

 Levi-Civita and Bisconcini. Levi-Civita 

 blazed the trail in the restricted problem 

 and found: an unique, invariant relation, 

 algebraical in the velocities, periodic and 

 uniform, which he developed in a power 

 series. It may be noted that simple modi- 

 fications of Levi-Civita 's analysis render it 

 immediately applicable to the restricted 

 problem of four bodies. There appears 

 again a single uniform periodic condition 

 for collisions of two of the bodies, and this 

 condition is algebraic in the velocities. 

 The result thus constitutes an exception to 

 Painleve 's theorem that when three of the 

 masses are different from zero the condi- 

 tions which must be satisfied in the w-body 

 problem in order that after a finite interval 

 of time two of the bodies may collide, can 

 not be algebraical conditions. In the gen- 

 eral three-body problem Bisconcini, fol- 

 lowing the route marked out by Levi-Civita 

 in the restricted problem, has arrived at 

 two distinct relations whose analytical form 

 he has determined. Bisconcini has thus 

 been able to characterize all the singular 

 motions of the system in which any two of 

 the bodies collide, and to determine the 

 analj^ical conditions under which we may 

 be certain that the motion will proceed 

 regTilarly. One of the assumptions made 

 by Bisconcini in the course of this work 

 has since been demonstrated by Sundman. 

 In a new elaboration of his original memoir 

 Levi-Civita has been able to extend certain 

 of his results to the astronomical, restricted 

 problem. Sundman has found the condi- 

 tion for the simultaneous collision of all 



three bodies to consist in a vanishing of all 

 three integrals of areas in the motion of 

 the bodies with respect to their common 

 center of gravity ; if the constants of areas 

 are not all zero, Sundman has assigned a 

 positive limit below which, of the three 

 distances, the greatest always remains so. 

 The same writer has announced the exten- 

 sion of his results to the n-hodj problem, 

 including explicit expressions for the co- 

 ordinates in the vicinity of equilibrium. 

 In the meantime Block has presented to the 

 Swedish Academy of Sciences a memoir in 

 which he has given the developments in 

 powers of the time in Sundman 's case of 

 collision; these power series contain terms 

 of three different forms in whose exponents 

 the masses of the bodies appear. The re- 

 cent memoirs of von Zeipel on intransitive 

 motion in the three-body problem and the 

 indeterminate singularities in the case of n 

 bodies are treated in the report reviewed 

 here. Mittag-Leffler is preparing a memoir 

 soon to be published in the Acta Mathe- 

 matica in which there will appear a digest 

 of Weierstrass's correspondence in its bear- 

 ing on the problem of three bodies. The 

 memoir will be concerned especially with 

 the relations of this correspondence to the 

 setting of the problem for the prize, offered 

 by the late King Oscar II., of Sweden; to 

 the report on which the award of the prize 

 was based; and to the recent work on the 

 singular trajectories of the general problem 

 of three bodies, and its resolution in power 

 series. 



V 



GENERALIZATIONS OF THE PROBLEM AND ITS 

 INVERSION 



During the period under discussion the 

 problem has been variously generalized. 

 Ebert has formulated an equivalent prob- 

 lem to that of 11 bodies, with an additional 

 integral; and a similar generalization has 

 been made by extending the Bour-Bertrand 



