REVIEWS 345 



The object of this book is to set forth in an elementary way the fundamental 

 properties of the elliptic functions. The whole work is divided into four parts, 

 consisling in all of twenty chapters : in the first part the Jacobian functions are 

 treated ; in the second the Weierstrassian functions ; in the third the general 

 principles of the subject from the point of view of the theory of functions ; and in 

 the fourth the Theta functions. In the first part, the first chapter deals with the 

 definitions of the functions sn, en, and dn, their elementary properties, their 

 real periods, their derivatives, their developments in series, their variations, and 

 their addition-formulae ; the second chapter with imaginary variables, the Jacobian 

 functions for their imaginary periods, and their zeros and poles ; the third chapter 

 with elliptic integrals ; the fourth chapter with numerical calculations of elliptic 

 integrals and functions ; and the fifth chapter with the direct calculation of 

 elliptic integrals of the first and second kinds. The only criticism which is 

 at all obvious in this very complete and thorough work seems to me to be in 

 the, to a student, not very clearly proved statement (p. 150) that if an elliptic 

 function were finite in an elementary parallelogram, it would be finite at all points 

 at a finite or i?ifinite distance, because of the double periodicity. No further 

 foundation of this statement is given, though it is evidently not necessarily the 

 case that a function which is finite at all finite points is finite everywhere. 



Philip E. B. Jourdain. 



A Treatise on the Analytical Dynamics of Particles and Rigid Bodies ; with 

 an Introduction to the Problem of Three Bodies. By E. T. Whittaker, 

 Hon. Sc.D. (Dubl.), F.R.S., Professor of Mathematics in the University of 

 Edinburgh. [Second edition. Pp. xii + 432.] (Cambridge : University 

 Press, 1917. Price 15^. net.) 

 The first edition of this excellent treatise was published in 1904, and there are 

 not very many important alterations in this second edition. The author says in the 

 preface to this edition that he has " endeavoured to give references to, and in 

 some cases accounts of, the numerous original researches in Dynamics which 

 have been published by various investigators since the first edition appeared." 

 The chief value of the book lies in the thorough, systematic, and modern treat- 

 ment of the integration and transformation of the dynamical equations. The 

 problems of dynamics are approached in a general and analytical way that 

 might, I think, be introduced with advantage at a somewhat earlier stage in our 

 teaching than is usually done. Much more light is thrown on the subject when 

 we inquire in general into the nature of those problems which makes them 

 solvable than if we merely collect individual examples of solvability, as if nature 

 were a huge examination paper ; and there is no need to spend quite such a long 

 time on what is called "elementary" work. The most noteworthy additions to 

 this edition are the new explanation of the contact-transformation theory of 

 dynamics and Hamilton's characteristic function (pp. 288-92), and the brief 

 account of Sundman's regularisation of the problem of three bodies (pp. 411-12). 

 Also there are some changes in the treatment of the motion of a body about a 

 fixed point under no forces (pp. 144-52) which arise from Prof. Whittaker's 

 opinion that the Jacobian elliptic functions are preferable to the Weierstrassian 

 ones in numerical calculations. The only marked criticism that seems to me 

 might be made on the new edition is that there is rather a noticeable — in view of 

 Prof. Whittaker's remarks in the preface — neglect of work published from 1904 to 

 1908 on the questions that arise in formulating Hamilton's principle and the 

 principle of least action for non-holonomic conditions and generalised co-ordinates. 



Philip E. B. Jourdain. 



