84 



SCIENCE 



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



while the rectilinear solution is always 

 unstable. 



Theoretical interest in the Lagrangian 

 solutions has been increased by Sundman's 

 theorem that the more nearly all three 

 bodies in the general problem tend to 

 collide simultaneously, the more nearly do 

 they tend to assume one or the other of 

 Lagrange's configurations; and on the 

 other hand practical interest in them has 

 been revived by the discovery of three 

 small planets, 1906 T.G., 1906 V.Y., 1907 

 X.M., near the equilateral triangular 

 points of the Sun-Jupiter- Asteroid system. 

 Linders has begun the investigation of the 

 motion of the first of these by starting 

 from a periodic solution of the differential 

 equations and developing the Jupiter per- 

 turbations from the osculating elements. 



Lehmann-Filhes, Hoppe and Dziobek 

 have generalized the exact solutions to 

 cases of more than three bodies placed on 

 a line or at the vertices of a regular poly- 

 gon or polyhedron, and isosceles tri- 

 angular solutions have been studied by 

 Fransen, Gorjatschew and Woronetz, 

 while Longley in an investigation of the 

 plane problem of invariable configuration 

 pays special attention to the rhombus. The 

 cases considered by Dziobek and Lehmann- 

 Filhes have been generalized by Pizzetti 

 in a direct study of the homographic mo- 

 tion of n bodies. Among the most in- 

 teresting extensions of Lagrange 's theorem 

 are those due to Banachiewitz and Moul- 

 ton. The former considers a non-equi- 

 lateral triangular system with fixed center 

 of gravity and under attractions according 

 to the inverse cube of the distance. He 

 finds a particular solution in which the 

 triangle rotates around the x-axis, its angles 

 remaining constant, and each point de- 

 scribing a curve on a cone of revolution 

 about the same axis which projects into a 

 spiral on the base of the cone. This is 

 the first case of an exact solution in which 



three finite bodies describe curves of double 

 curvature. Moulton's case is that of the 

 four-body problem consisting of three 

 arbitrary masses, in motion according to 

 either of Lagrange's solutions, and an in- 

 finitesimal body; there are eighteen solu- 

 tions of arbitrary period in which the 

 finite bodies lie on a line, and ten in 

 which they are at the vertices of an equi- 

 lateral triangle. Periodic solutions an- 

 alogous to those in the restricted three- 

 body problem have been constructed for 

 Moulton's problem. 



The method of Lagrange's memoir has 

 been extended to the four-body problem by 

 Seydler and more recently by "Woronetz; 

 the latter has pointed out particular solu- 

 tions in which three of the bodies are 

 equal ; these solutions are given by quadra- 

 tures if the law of force is inversely as the 

 cube of the distance and are capable of 

 direct extension to the case of any number 

 of bodies. 



Ill 



PERIODIC SOLUTIONS AND THEIE APPLICA- 

 TIONS 



The Lagrangian solutions remained the 

 only known periodic solution of the prob- 

 lem of three bodies for one hundred and 

 five years until 1877, when Hill, in his 

 epoch-making researches on the lunar 

 theory, demonstrated the existence of a 

 periodic solution which could serve as the 

 starting point for a study of the moon's 

 orbit. With these memoirs he broke 

 ground for the erection of the new science 

 of dynamical astronomy whose mathe- 

 matical foundations were laid broad and 

 deep by Poincare. Up to the time when 

 HiU 's work appeared, mathematical astron- 

 omers were accustomed to assume a solu- 

 tion of the problem of two bodies as a 

 first approximation in the lunar theory; 

 which intermediate orbit includes none of 

 the inequalities due to the sun's disturbing 



