REVIEWS 



MATHEMATICS 



Differential Equations. By H. Bateman, Ph.D., M.A., Lecturer in Applied 

 Mathematics, Johns Hopkins University, Baltimore ; formerly Fellow of 

 Trinity College, Cambridge ; and Reader in Mathematical Physics in the 

 University of Manchester. [Pp. xii + 306.] (London and New York : 

 Longmans, Green & Co., 1918.) Price 16s. net. 



An excellent addition to the excellent "Longmans' Modern Mathematical Series." 

 The technic of solving differential equations began to appear very early in the 

 history of the calculus, because many questions in mechanics and geometry could 

 not immediately be expressed as explicit or implicit relations between the 

 variables alone, but only in the form of relations between these variables and 

 their differentials, which, in many cases, could be solved so as to express some — 

 but not all — of the properties of such relations as are indicated above. This is 

 the point from which the present book starts (p. 1), and the book is principally 

 concerned with methods of actual solution of differential equations, as opposed to 

 more theoretical treatises. In fact, general theorems on existence of solutions are 

 not touched until disintegration by series is considered (pp. 223, 245). 



" In writing this book," says the author (p. v), " I have endeavoured to supply 

 some elementary material suitable for the needs of students who are studying the 

 subject for the first time, and also some more advanced work which may be 

 useful to men who are interested more in physical mathematics than in the 

 developments of differential geometry and the theory of functions. The chapters 

 on partial differential equations have consequently been devoted almost entirely 

 to the discussion of linear equations." After a first (preliminary) chapter on 

 differential equations and the nature of their solutions, in which are treated 

 discontinuous solutions and Green's function for one dimension, precedent is 

 slightly departed from in that, instead of beginning with the usual forms of 

 equations which can be solved very easily, the second chapter is on integrating 

 factors, and the third chapter is on transformations of given equations into others 

 which can be integrated directly or which present certain other advantages. The 

 fourth chapter is on geometrical applications, and the fifth is on differential 

 equations with particular solutions of a specified type. The sixth chapter is on 

 partial differential equations, and the treatment of special solutions bears some 

 resemblance to a treatment lately given by Prof. M. J. M. Hill (cf. Science 

 PROGRESS, 1918, 12, 548), though Dr. Bateman's treatment is independent of 

 Prof. Hill's paper (p. v). The seventh chapter is on total differential equations, 

 the eighth is on partial differential equations of the second order, and both of 

 these chapters contain a few results which seem new (p. v). The remaining three 

 chapters are on integration in series, the solution of linear differential equations 

 by means of definite integrals, and the mechanical integration of differential 

 equations. 



This book is an admirable one ; the examples are particularly interesting — 



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