GEOLOGY — MOULTON. 223 



Linear Differential Equations with Periodic Coefficients: 



This paper was prepared with the collaboration of Dr. W. D. MacMillan, 

 of the University of Chicago. It takes up first in a new way the proof of the 

 analytic character of the solutions in the general case of a simultaneous set 

 of equations of any order. Then it treats those equations whose coefficients 

 have the property of being expansible as power series in certain parameters, 

 and exhibits not only the character of the solutions with respect to these 

 parameters, but shows how actually to construct them by relatively simple 

 and convenient processes. These are the equations which arise in celestial 

 mechanics and their solutions in all cases are now at hand. The most useful 

 cases for equations with right members are also treated. This paper is being 

 published by the American Journal of Mathematics. 



Problem of the Spherical Pendulum from the Standpoint of Periodic Orbits: 



The problem of the spherical pendulum falls in the class of those which 

 can be treated by the methods of periodic orbits. The solution of the ^-equa- 

 tion leads to elliptic functions which are obtained here expanded as power 

 series in their modulus directly from the second-order differential equation. 

 The method is applicable, with slight modifications, to hyperelliptic func- 

 tions. After the ^-equation has been solved the x and y equations become 

 linear of the second order with periodic coefficients which are expansible as 

 power series in the modulus of the elliptic functions. The solutions of these 

 equations are found. After the properties of the solutions have been derived 

 from the original differential equations, the remarkable fact is shown that all 

 the coefficients can be obtained from the integral relations which hold among 

 the coordinates. Thus a second independent method of computing them is 

 given. This paper is being submitted to the Rendiconti di Palermo for pub- 

 lication. 



A Certain Class of Oscillating Satellites: 



In this paper two finite masses are supposed to be describing undisturbed 

 elliptical orbits, and an infinitesimal mass is moving near one of the La- 

 grangian centers of libration. The conditions under which its oscillations 

 can be periodic are determined, and a method of finding these solutions is 

 given. The coordinates are expansible as sums of fractions of a parameter, 

 of which they are discontinuous functions. 



The work under preparation is : 



The first three subjects described in Year Book No. 8, p. 225. 



Continuation of the work on periodic orbits for a second volume. 



The section on cosmogony for Encyclopaedic der Mathematische Wissenschaften. 



