222 REPORTS ON INVESTIGATIONS AND PROJECTS. 



GEOLOGY. 



Chamberlin, T. C, University of Chicago, Chicago, Illinois. Grant No. 571. 

 Study of fundamental problems of geology. (For previous reports see 

 Year Books Nos. 2-8, inclusive.) $4,000 



The year has been spent mainly on special inquiries bearing on the sources 

 and the methods of supply and of depletion of atmospheric material and on 

 the regulative factors that enter into the maintenance of the equilibrium of 

 the atmosphere and into the limitation of its oscillations, as these are implied 

 by the climatic data of the geologic record. A paper on the secular mainte- 

 nance of the atmosphere, embodying these results, has been in course of 

 preparation during the last quarter of the year. 



Moulton, F. R., University of Chicago, Chicago, Illinois. Grant No. 627. 



Continuation of investigations relating to the planetesimal hypothesis. 



(For previous reports see Year Books Nos. 4, 5, and 8.) $2,000 



The work finished during the year and now being published or ready for 



publication is : 



Periodic Orbits (vol. I) : 



For description see Year Book No. 8. 



The Straight-Line Solutions of the Problem of Bodies: 



In this paper it is proved that it is possible to arrange any n positive finite 

 masses on a straight line in precisely y 2 n ! different ways, so that, under 

 proper initial projections, they will always remain collinear. The orbits of 

 the masses are similar conic sections having the center of mass of the whole 

 system as a focus. This is the complete generalization of Lagrange's results 

 for the Problem of Three Bodies. 



The related problem is solved of determining, when possible, n masses such 

 that if they are placed at n arbitrary collinear points they will, under proper 

 initial projection, always remain in a straight line. If n is even and the 

 linear dimensions of the orbits are given, it is proved that the n masses are 

 in general uniquely determined ; and that if n is odd the coordinates of the n 

 points must satisfy one algebraic relation, after which, choosing any one of 

 the masses arbitrarily, the remaining n — 1 are uniquely determined. This 

 paper is in type for the Annals of Mathematics. 



The Singidarities of the Two-Body Problem for Real Initial Conditions: 



From the standpoint of analysis, the most important properties of a func- 

 tion are the location and character of its singularities. They determine the 

 character of its expansibility at every point. This paper makes a complete 

 discussion for the problem of two bodies for all real initial conditions. It 

 has been submitted to the Transactions of the American Mathematical So- 

 ciety for publication. 



