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elliptic integrals of the first kind to Legendre's normal form, and the fourth chapter 

 is on numerical computation of the elliptic integrals of the first and second kinds, 

 and on Landen's transformations. The fifth chapter is concerned with various 

 examples and problems — the rectification of the lemniscate and the ellipse, the 

 motion of a pendulum, and so on. The sixth chapter gives five-place tables, from 

 Levy's Thdorie, of integrals of the first and second kind. There is a short account 

 of the history of the subject in the introduction (pp. 6-8), but, of course, in such 

 a small space the account is not of much interest to a student. This book is of 

 value to those students who have a taste for the rather complicated technical 

 work of calculation involved in this " higher trigonometry" : it seems to me much 

 the best plan to introduce students to the study of elliptic functions in the way 

 followed in this book. Philip E. B. Jourdain. 



Lecons sur les Methodes de Sturm dans la theorie des equations differentielles 

 lineaires, et lenrs developpements modernes, professees a la Sorbonne 

 en 1913-1914. By Maxime Bocher, Professor at Harvard University 

 and (temporarily) at the University of Paris. Collected and edited by 

 GASTON Julia. [Pp. vi + 118, with 8 figures.] (Paris: Gauthier-Villars 

 et Cie., 191 7. Price 5 fr.) 

 THIS book is a very well written introduction to the interesting methods first 

 developed by Sturm in a memoir published in 1836 for the discussion of what are 

 now known as boundary value problems with ordinary differential equations. Of 

 course the question of boundary values first comes before a student, as it did in 

 history, in discussing the partial differential equations of mathematical physics, but 

 it is well known that such discussions can often be reduced to the study of 

 certain ordinary differential equations of the second order. The third chapter of 

 the present treatise is devoted to a study of the results given by Sturm in his first 

 memoir ; his first great theorem (pp. 45-6) is that the zeros of the real independent 

 solutions of a linear and homogeneous differential equation of the second order 

 with real coefficients are mutually separated from one another. The first chapter 

 is a treatment of the theorems on the existence of solutions of ordinary real and 

 linear differential equations of the second order ; the second chapter is on the 

 analogies of linear differential systems with linear algebraic systems — Sturm, 

 like many before him, having regarded differential equations as limits of difference 

 equations ; the fourth chapter is on the characteristic functions and their zeros in 

 two problems which go beyond the problems of Sturm treated in the third chapter, 

 and here there is some account of the important work of Klein (1881) ; and the 

 fifth chapter is on those properties of Green's functions which can be extended to 

 ordinary linear differential equations and their applications. It should be added 

 that this volume is one of M. E. Borel's famous series of monographs on the theory 

 of functions. Philip E. B. Jourdain. 



Lee, cms snr les Fonctions Elliptiques en vue de leurs applications. By R. de 

 MONTESSUS DE Ballore. Cours libre professe a la Faculte des Sciences 

 de Paris. [Pp. x + 267.] (Paris: Gauthier-Villars et Cie., 1917. Price 12 fr.) 

 There have been lately many indications of a growing feeling that, for purposes 

 of application to mathematical physics, the elliptic functions of Jacobi are far more 

 convenient to deal with than those of Weierstrass. In fact, it is possible to begin 

 the study of elliptic functions as soon as the principles of the integral calculus 

 have been mastered, and this study is necessary, for at each step in applied 

 mathematics we meet elliptic functions, and it seems a mistake to consign the 

 theory of such functions to the "higher" branches of analysis. 



