326 SCIENCE PROGRESS 



isosceles triangle solutions of the three-body problem : if a third body is 

 started from the centre of gravity of two equal masses with initial conditions 

 so chosen that it moves in a straight line and remains equidistant from the 

 other bodies, periodic orbits may exist whether the third body is infinitesimal 

 or finite in mass and whether the finite bodies move in a circle or in ellipses. 

 All these cases are fully discussed. Chapters XI and XII deal with related 

 subjects : the periodic orbits of infinitesimal satelhtes and inferior planets 

 and of superior planets respectively. The latter case has no direct bearing 

 upon the problems presented by the solar system, but, in the former case, the 

 results are of direct practical application, particularly in the lunar theory. 

 Analogous problems have been treated by Hill and Brown. It is shown that 

 there are three famiUes of satellites and of inferior planets whose motion is 

 direct and an equal number with retrograde motion. Darwin, for a particular 

 value of the mass-ratio, found the three satellite orbits but only one planetary 

 orbit. The methods used in these two chapters are very general, and are 

 applicable to any ratio of masses of the finite bodies. In Chapter XIII, the 

 periodic orbits of a particle subject to the attraction of n-spheres having 

 prescribed motions are discussed. Chapter XIV deals with a problem which 

 is similar to that presented by the satellite systems of Jupiter and Saturn, 

 the periodic orbits of k finite bodies revolving about a relatively large central 

 mass, together with many related questions. Chapter XV contains a discus- 

 sion of limiting cases of periodic orbits, viz. closed orbits of ejection, which pass 

 through one of the masses. This is followed by a final chapter, in which an 

 attempt is made to trace out the evolution of periodic orbits as the parameters 

 upon which they depend are varied. An enormous amount of computation 

 was involved in the preparation of this chapter. The non-existence of 

 isolated periodic orbits is shown, and much light is thrown upon the inter- 

 relationship of various families of periodic orbits although the discussion is 

 admittedly in certain respects incomplete. 



The labours of Prof. Moulton and of his collaborators have brought to 

 light many new famiUes of periodic orbits, which have been discussed 

 with greater generality and rigour than previously. This volume is a fitting 

 monument to their work ; it will prove invaluable to all future students of the 

 subject. 



H. S. J. 



PHYSICS 



The Theory oJ Relativity. By Robert D. Carmichael, Professor of 

 Mathematics in University of Ilhnois. Second Edition. [Pp. 112, 

 with 8 figures.] (New York : John Wiley & Sons. London : Chap- 

 man & HaU, 1920. Price 8s. 6d. net.) 



This volume forms one of an excellent series of Mathematical Monographs 

 by American authors, edited by Mansfield Merriman and Robert S. Wood- 

 ward. The first edition was published in 1913, but the successful formulation 

 by Einstein of a general theory of relativity embracing gravitation has called 

 for a second edition in which an additional chapter has been added dealing 

 with this wider theory, the treatment of the older theory remaining precisely 

 as in the first edition. 



Being a monograph, the subject is necessarily treated somewhat briefly. 

 The author has nevertheless endeavoured throughout to distinguish between 

 what is based upon experience and what is postulated in the theory and to 

 examine to what extent the deductions from the postulates are compatible 

 with experiment. It is precisely in this respect that some accounts of 

 the theory are defective. For a logical, concise treatment of the special 

 theory requiring a relatively slight acquaintance with mathematics the first 

 six chapters can be strongly recommended. 



The account of the generalised theory given in the last chapter is based 



