NA TURE 



31. 



THURSDAY, AUGUST i, 1895. 



LINEAR DIFFERENTIAL E<2UATI0NS. 

 Handbuch der Theorie dcr linearen Differentialgkich- 

 un^cn. \'on Prof. Dr. Ludwig .Schlesinger, Privat- 

 docenten an der Universitiit zu Berlin. Erster Band. 

 (Leipzig: Teubner, 1895.) 



DE MORGAN is reported to have said of the subject of 

 differential equations, that it illustrated the proverb 

 that he who hides knows how to find. This was true 

 enough at a time when the sole aim of the analyst was 

 to "solve" differential equations by reducing them to 

 quadratures, or to construct ingenious puzzles for the 

 benefit of undergraduates. Integration by series was 

 kno\vn, of course ; but this was regarded as a mean 

 device, useful indeed for purposes of calculation, especi- 

 ally to the physicist, but unworthy of the serious attention 

 of the pure mathematician. 



A new era began with the foundation of what is 

 now called function-theory by Cauchy, Riemann, and 

 Weierstrass. The study and classification of functions 

 according to their essential properties, as distinguished 

 from the accidents of their analytical forms, soon led 

 to a complete revolution in the theory of differential equa- 

 tions. It became evident that the real question raised 

 by a differential equation is not whether a solution, 

 assumed to exist, can be expressed by means of known 

 functions, or integrals of known functions, but in the 

 first place whether a given differential equation does really 

 suffice for the definition of a function of the independent 

 variable (or variables), and, if so, what are the character- 

 istic properties of the function thus defined. Few things 

 in the history of mathematics are more remarkable than 

 the de\ elopments to which this change of view has given 

 rise. Leaving out of account the theorj' of partial 

 differential equations, which is still beset with many and 

 serious difiicultics, it is not too much to say th.at in the 

 course of less than half a century the theory of ordinary 

 linear differential equations alone has attained a degree 

 of extent and importance which makes it comparable 

 with ahnost any of the main branches of analysis. 



The landmarks of the new departure are the memoir 

 of Briot and Bouquet in the Journal dc PEcole Poly- 

 techniiiiie (cap. 36), Riemann's paper on the generalised 

 hypergeometric series, and Fuchs's memoir in the sixty- 

 sixth volume of CreUe's Journal. Since the publication 

 of this last work, more especially, the progress made 

 has been exceedingly rapid : the general principles of 

 the subjects have been permanently established, so as 

 already to admit of methodical treatment, and numerous 

 particular applications, all of great interest and beauty, 

 have attracted and continue to invite the attention of 

 mathematical explorers. Thus there is the problem of 

 discovering whether a given equation has an algebraic 

 integral, and, if so, of finding it ; there is the theory of 

 equations with doubly periodic coefficients ; and there is 

 the theory of differential invariants. Each of tlicse is 

 associated with some of the most brilliant discoveries 

 of modern analysis, and each offers abundant oppor- 

 tunity for further research. 

 The wide extent of the subject, and the immense 

 NO. 1344, VOL. 52] 



number of memoirs relating to it, have created an urgent 

 need for systematic treatises to serve as an introduction 

 to the theory, and presenting its main outlines in a proper 

 perspective. Fortunately this want seems likely to be 

 supplied before long ; various excellent works, dealing 

 wholly or in part with linear differential equations, have 

 recently appeared or are in course of publication, and 

 among these the book now under review will take an 

 honourable place. 



Dr. Schlesinger's work, to be completed in two 

 volumes, is intended to give a coherent and comprehensive 

 account of the theory in the light of its most recent 

 developments. This first volume is divided into eight 

 sections, exclusive of two introductory chapters, one 

 historical, the other treating of the existence of an 

 integral, and the general nature of the singular points. 

 Of the eight sections, the first contains the first principles 

 of the theory, mostly after Fuchs ; the second discusses 

 systems of independent integrals, reduction when par- 

 ticular integrals are known, Lagrange's adjoint equation, 

 non-homogeneous equations, and Frobenius's theorems 

 on irreducibility ; the third relates to the funda- 

 mental equation ; the fourth to unessential singular 

 points ; the fifth to equations of the " Fuchsian " class, 

 that is to say, of which the coefficients are rational 

 functions oi x and all the integrals are regular ; the si.xth 

 treats of the development of integrals within a circular 

 annulus ; and, finally, the seventh and eighth contain the 

 general theory of equations with rational coeflficients. 



The treatment is entirely analytical, and is based 

 principally on the methods of Weierstrass as expounded 

 by Fuchs, Frobenius, Hamburger and others ; thus the 

 integrals are obtained in the form of power-series vahd 

 within a certain region of the plane of the complex 

 variable, and no use is made of geometrical diagrams 

 such as those employed by Schwarz, Klein, and Goursat. 

 Moreover, except in the fifth section, which contains a 

 brief discussion of Riemann's P-function and of the 

 hypergeometric series, the author confines himself to the 

 general theory, and does not consider special cases, or 

 particular applications. The demonstrations, for the 

 most part, are concise, and free use is made of the sign 

 of summation and double suffixes. For these reasons 

 the book is perhaps hardly suitable for those who are 

 approaching the subject for the first time ; but any one 

 who has read, let us say, Goursat's thesis on the hyper- 

 geometric series, or Klein's lectures on linear differential 

 equations of the second order, and is moderately familiar 

 with the Weierstrassian function-theory, will be able to 

 consult it with advantage. To those who are engaged in 

 research. Dr. Schlesinger's treatise will be of great value, 

 because those parts of the subject which are included 

 within the author's plan are discussed with sufficient 

 thoroughness, with a consistent notation, and in logical 

 order ; while the analytical table of contents gives 

 references to the original sources in direct connection 

 with the articles of the book which are based upon them. 

 It is rather a pity, by-the-by, that the dates ha\e not 

 always been gi\en in these references ; the reader may 

 very possibly wish to know the date of a paper, and not 

 be able to consult the volume of the journal in wliich it 

 appeared. 



Mathematicians will look forward with interest to 



