Januaet 16, 1920] 



SCIENCE 



53 



view formulated above. For, in a broad 

 sense, tbe question is that of determining the 

 general character of the limitations upon the 

 possible variations of the coordinates in dy- 

 namical problems. 



We wish to mention briefly four important 

 steps in advance in this direction. 



The first is due to Hill who showed in his 

 paper that, in the restricted problem of three 

 bodies, with constants so chosen as to give the 

 best approximation for the lunar theory, the 

 Moon remains within a certain region about 

 the Earth, not extending to the Sun. In fact 

 here there is an integral yielding the squared 

 relative velocity as a fimction of position, and 

 the velocity is imaginary outside of this 

 region. 



In his turn, Poincare showed that stability 

 exists in another sense, namely for arbitrary 

 values of the coordinates and velocities there 

 exist nearby possible orbits of the Moon 

 which take on infinitely often approximately 

 the same set of values. His reasoning is ex- 

 tremely simple, and is founded on a hydro- 

 dynamic interpretation in which the orbits 

 appear as the stream lines of a three-dimen- 

 sional incompressible fluid of finite volume in 

 steady motion. A moving molecule of such a 

 fluid must indefinitely often partially re- 

 occupy its original position with indefinite 

 lapse of time, and this fact yields the stated 

 conclusion. 



In 1901 under the same conditions Levi- 

 Civita proved that, if the mean motions of 

 the Sun and Moon about the Earth are com- 

 mensurable, instability exists in the following 

 sense: orbits as near as desired to the funda- 

 mental periodic lunar orbit will vary from 

 that periodic orbit by an assignable amount 

 after su£B.cient lapse of time. This result, 

 which is to be anticipated from the physical 

 point of view, makes it highly probable that 

 instability exists in the incommensurable case 

 also. 



These three results refer to the restricted 

 problem of three bodies. 



Finally there is Sundman's remarkable 

 work on the unrestricted problem contained 

 in his papers of 1912 and of earlier date. 

 Lagrange had proved that if a certain energy 



constant is negative, the sum of the mutual 

 distances of the three bodies becomes infinite. 

 Sundman showed that, even if this constant 

 is positive, the sum of the three mutual dis- 

 tances always exceeds a definite positive quan- 

 tity, at least if the motion is not essentially 

 in a single plane. Thus he incidentally veri- 

 fied a conjecture of Weierstrass that the 

 three bodies can never collide simultaneously. 

 These and other results seem to me to render 

 it probable that in general the sum of the 

 three distances increases indefinitely. Thus, 

 if this conjecture holds, in that approxima- 

 tion where the Earth, Sun and Moon are 

 taken as three particles, the Earth and Moon 

 remain near each other but recede from the 

 Sun indefinitely. The situation is worthy of 

 the attention of those interested in astronomy 

 and in atomic physics. 



As we have formulated the concept of 

 stability, it is essentially that of a permanent 

 inequality restricting the coordinates. We 

 may call a dynamical system transitive in a 

 domain under consideration if motions can be 

 found arbitrarily near any one state of motion 

 of the domain at a particular time which pass 

 later arbitrarily near any other given state. 

 In such a domain there is instability. If we 

 employ the hydrodynamic interpretation used 

 above, the molecule of fluid will diffuse 

 throughout the corresponding volume in the 

 transitive case, and will diffuse only partially 

 or not at all in the intransitive case. The 

 geodesies on surfaces of negative curvature, 

 treated by Hadamard in 1898, furnish a 

 simple illustration of a transitive system, 

 while the integrable problem of two bodies 

 yields an intransitive system. Probably only 

 under very special conditions does intransi- 

 tivity arise. 



It is an outstanding problem of dynamics 

 to determine the character of the domains 

 within which a given dynamical system is 

 transitive. 



A less difficult subject than that of stability 

 is presented by the singularities of the 

 motions such as arise in the problem of three 

 bodies at collision. The work of Levi-Civita 

 and Sundman especially has shown that the 

 singularities can frequently be eliminated by 



