MATHEMATICAL PHYSICS. 265 



Certain reviews and popular or semipopular articles have been published. 

 The following investigations have been completed during the last year, and 

 are now in type, though they have not yet been published : 



(i) Closed orbits of ejection and related periodic orbits. Painleve's Theorem. Pro- 

 ceedings London Mathematical Society, Series 11, vol. 11, pp. 367-397. 



In this paper it is shown that there exist two infinite systems of orbits of 

 ejection of an infinitesimal body having the property that they are also orbits 

 of collision. The direction of ejection is either toward or from the second 

 finite body, and the direction of collision is opposite to that of ejection. The 

 closed orbits of ejection are not periodic, but they are important in the sub- 

 ject of periodic orbits because, as is shown in the paper, each of them is the 

 limit of two families of periodic orbits. In Painleve's Stockholm lectures on 

 the theory of differential equations it was stated, as a conjecture, that the 

 coordinates and components of velocity of the infinitesimal body must sat- 

 isfy one analytic condition in order that its orbit should be one passing 

 through a finite body. It is proved in this paper that when one of the finite 

 bodies is small, a point and a speed may be chosen arbitrarily, and that then 

 a direction of motion can be determined as an analytic function of these ar- 

 bitraries so that the infinitesimal body will pass the chosen point with the 

 chosen speed, so that its orbit will be either one of ejection or one of collision. 



(2) Oscillating satellites in the problem of three bodies. The Mathematische Annalen, 



39 pages. 



This paper contains a general investigation of oscillating satellites, in both 

 two and three dimensions, about the collinear equilibrium points when one 

 of the masses is infinitesimal. When the finite bodies move in circular orbits, 

 six families of periodic oscillating satellite orbits are shown to exist, their 

 properties are established, and practical means are given for constructing 

 them. When the finite bodies move in elliptical orbits, the periodic orbits 

 are closed only after many revolutions, and for a given period (not fully 

 arbitrary) there are twelve geometrically distinct orbits. The discussion in- 

 volves the treatment of an infinite series of simultaneous linear differential 

 equations having periodic coefficients, and right members which are sums of 

 exponentials multiplied by periodic functions. 



(3) Relations among families of periodic orbits in the restricted problem of three 



bodies. Proceedings International Congress of Mathematics. 



This paper gives a synthesis of the retrograde periodic orbits in the prob- 

 lem of three bodies. It shows the connections between the oscillating satel- 

 lites and the closed orbits of ejection, of these and Poincare's solutions of 

 the deuxicme sorte, and of these with his solutions of the premiere sorte, and 

 of all with his solutions of the deuxieme genre. 



A general synthesis of periodic orbits is almost finished, and will appear 

 as a chapter in Publication i6i of the Carnegie Institution of Washington, 

 now in press. It will include the paper presented at the International Con- 

 gress of Mathematics, and the corresponding discussion of the direct orbits. 



