REVIEWS 325 



ASTRONOMY 



Periodic Orbits. By F. R. Moulton, in collaboration with D. Buchanan, 

 T. Buck, F. L. Griffin, W. R. Longley, and W. D. MacMillan. 

 [Pp. XV +524.] (Washington: Carnegie Institution, 1920.) 

 This volume contains in collected form the researches on the subject of periodic 

 orbits carried on from 1900 onwards by Prof. F. R. Moulton or by students 

 who made their doctorates under his direction. The substance of many of 

 the chapters had been published previously in various mathematical or 

 astronomical journals, but the Carnegie Institution has rendered a valuable 

 service in publishing the investigations in collected form and in extenso. A 

 tribute should be paid to the quality of the paper and of the printing ; the 

 type used is very clear, and great care has been taken in the printing of compli- 

 cated mathematical formulas and equations. A handsome volume is the 

 result. 



The distinguishing feature of the treatment throughout is its mathemati- 

 cal rigour. The validity of the methods used for the solution of differential 

 equations is established and the consequence of solutions in infinite series is 

 proved. In cases where several types of periodic orbits are possible the 

 existence of the several types is formally proved and the method of obtaining 

 the orbits is developed. And here it must be mentioned that careful regard 

 has been had to practical requirements, so that the developments are always 

 in a form applicable to practical problems in celestial mechanics. 



In Chapter I methods of solution of certain types of differential equations, 



required in the subsequent chapters, are explained and the convergence of 



the solutions is examined. Chapter II treats of questions dealing with 



elliptic motion which are classic in celestial mechanics, but, instead of following 



the ordinary treatment, the periodicity of the motion is established and 



direct solutions in terms of the time as variable parameter are constructed 



by the methods developed in the preceding chapters. This application of 



these methods at once indicates their analytical power. Chapter III deals 



with the spherical pendulum, the same methods being used : it contains, 



amongst other things, a new and rigorous treatment of Hill's differential 



equation with periodic co-efQeients. In Chapter IV periodic orbits about 



an oblate spheroid are discussed. Such orbits are not in general closed 



geometrically, but, considering the orbit in a revolving meridian plane passing 



through the particle, several classes of closed periodic orbits can be obtained ; 



if the period of rotation of the line of nodes is commensurable with the period 



of motion in the revolving plane, the orbit is also closed in space. This 



problem has an apphcation in the case of an oblate planet, such as Jupiter. 



Chapters V, VI, and VII deal with the periodic orbits of satellites oscillating 



about the straight line equilibrium points. Lagrange showed that if two finite 



spherical bodies revolve about their common centre of mass in circles, there 



are three points in the line of the masses such that if small masses are placed 



at them and projected so as to be instantaneously at rest relatively to the 



revolving system, they will always remain fixed relatively to it. Moulton 



shows that, if one of the small bodies is given a slight displacement, it may, 



under certain conditions, revolve in the vicinity of the equilibrium point in 



an orbit closed relatively to the revolving system. The existence of such 



orbits and their direct construction by two different methods are detailed : 



the corresponding problems are then solved for the case when the finite masses 



describe elliptical instead of circular orbits. In Chapter VIII generalisations 



of these problems to the case of n bodies are given : it is shown that there are 



\n ! straight-Line solutions, such that under proper initial projections the 



bodies will remain always collinear and that to each one of these solutions 



there are (w + i) points of libration near which oscillating satellite orbits 



are possible. Chapter IX deals with corresponding periodic orbits near the 



Lagrangian equilateral triangle points. In Chapter X are discussed the 



