January 15, 1909] 



SCIENCE 



83 



mentum, and the integral of energy. The 

 question of further progress in this reduc- 

 tion is vitally related to the non-existence 

 theorems of Bruns, Poincare and Painleve. 

 Bruns demonstrated that the w-body prob- 

 lem admits of no algebraical integral other 

 than the ten classic ones, and Poincare 

 proved the non-existence of any other uni- 

 form analytical integral. A strikingly in- 

 structive example illustrating these non- 

 existence theorems has been given by 

 Perchot and Ebert. Painleve has general- 

 ized Bruns 's theorem by showing that, in 

 addition to the classical integrals of energy 

 and momentum, there exists neither in- 

 tegral nor integral equation algebraic in 

 the velocities, and the theorem of Poincare, 

 by proving that there exists no new 

 analytical integral unifonn with respect to 

 the velocities. Grave showed that the 

 three-body problem under forces varying 

 as any function of the distance possesses 

 no new integral independent of the law of 

 attraction, and this theorem has been gen- 

 eralized for the ?i-body problem. Bohlin 

 has very recently added to the non-exist- 

 ence theorems by demonstrating that the 

 mutual distances in the problem of three 

 bodies can not be expressed as the roots 

 of an algebraical equation of the fifth 

 degree with transcendental coefficients. 



II 



PAETICULAR SOLUTIONS AND THEIR 

 GENERALIZATIONS 



In 1772 the prize of the Academic 

 Royale des Sciences de Paris was awarded 

 to Lagrange for an "Essai sur le Prob- 

 leme des Trois Corps. ' ' In this celebrated 

 memoir Lagrange "shows that the cUn- 

 plete solution of the problem requires only 

 that we know at each instance the sides 

 of the triangle formed by the three bodies ; 

 the coordinates of each may then be deter- 

 mined without difficulty. As for the solu- 

 tion of the triangle, it depends upon three 



differential equations, of which two are of 

 the second order, the third of order three. ' ' 

 He determined all the solutions of the 

 problem in which the ratios of the mutual 

 distances of the bodies remain constant. 

 In one of the two distinct configurations 

 the bodies are always at the vertices of an 

 equilateral triangle; in the other they lie 

 always on a straight line. In both of 

 these cases the motion of each body rela- 

 tive to either of the others is the elliptic 

 motion of the two-body problem. Tscherny 

 has constructed these solutions geomet- 

 rically; he has also shown that the only 

 cases of the three-body problem for which 

 known mathematical and mechanical means 

 suffice are those which reduce to the prob- 

 lem of two bodies. Lagrange's solutions 

 were originally discovered in his problem 

 of the mutual distances; the latter, called 

 by Hesse the reduced problem, has re- 

 cently assumed a new form under Char- 

 lier's treatment, in which the mutual dis- 

 tances are replaced by the distances from 

 the center of gravity. From Lagrange's 

 discussion certain imaginary considera- 

 tions were omitted; Whittemore has filled 

 this gap, but the completed discussion 

 yields no other real solution. The equi- 

 lateral triangular solution is possible for 

 all distributions of the masses; their dis- 

 tribution on the straight line is defined by 

 the real positive root of a certain quintic 

 equation; Frederigo has given a new 

 derivation of this equation and Bohlin has 

 formulated four developments, of which 

 three represent the roots of the quintic in 

 three distinct domains, and the fourth for 

 an isolated value. The question of the 

 stability of the solutions furnishes Levi- 

 Civita an example of his theory of station- 

 ary motion in which reappear the results 

 of Liouville and Routh, namely, the tri- 

 angular solution is stable if 



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