264 REPORTS ON INVESTIGATIONS AND PROJECTS. 



Morley, Frank, Johns Hopkins University, Baltimore, Maryland. Grant 



No. 755, allotted December 15, 1911. Application of Cremona groups 



to the solution of algebraic equations. (For previous reports see Year 



Books Nos. 9, 10.) $1,200 



The general trend of our researches for the year was determined by 



questions arising from the rational planar quintic curve. 



(i) The study of this curve suggested a theorem with regard to any 

 planar quintic, namely, that the contact conies touch, by threes, certain lines 

 (Johns Hopkins Circular, Feb. 1912). Professor Coble was led thereby to 

 investigate the general question of the grouping of contact curves by the 

 method of finite geometry. The results were indicated in the above circular 

 and are embodied in a memoir submitted (June 1912) to the Transactions. 

 (2) The general properties of an involution form were discussed by 

 Coble (Am. Jour., vols, xxxi and xxxii). A construction for the so-called 

 fundamental involution was obtained in a memoir which will appear in the 

 Proceedings of the Cambridge International Congress, which will include 

 also a solution of an important problem of enumeration. The difficulty of 

 this enumeration by methods hitherto available seems to show the desira- 

 bility of overhauling the general subject of the order of restricted systems 

 of equations. 



MATHEMATICAL PHYSICS. 



Moulton, F. R., University of Chicago, Chicago, Illinois. Grant No. 770, 

 allotted December 15, 1911. Investigations in cosmogony and celestial 

 mechanics. (For previous reports see Year Books 4, 5, 8-10.) $2,000 



During the past year the following papers have been published: 



(1) On certain expansions of elliptic, hyperelliptic, and related periodic functions. 



American Journal of Mathematics, vol. 34, pp. 177-202. 



In addition to its more general features, this paper contains new and very 

 convenient expansions of the Legendre Elliptic Functions. 



(2) The problem of the spherical pendulum from the standpoint of periodic solutions. 



Rendiconti del Circolo Matematico di Palermo, vol. 32, pp. 338-364. 



Among the interesting features of this paper is a new treatment of Hill's 

 linear differential equations with periodic coefficients, and the use of the 

 integral relations for the determinations of the coefficients of the solutions. 



(3) A class of periodic orbits of superior planets. Transactions of the American 



Mathematical Society, vol. 13, pp. 96-108. 



This paper proves the existence of, and gives practical means of construct- 

 ing, periodic orbits of an infinitesimal body revolving around two finite bodies 

 which move in circular orbits. The orbits in question are those which are 

 closed after one synodic revolution. There are three classes of orbits in 

 which the motion is direct, with respect to fixed axes, and three in which it 

 is retrograde ; but when the orbits are very large, only one class for motion 

 in each direction is real. 



