392 



SCIENCE. 



[Vol. XIV. No. 357 



form ; " that is to say, by means of the elementary functions of 

 analysis. But though the importance of this problem for practical 

 purposes must be acknowledged, the problem itself, understood in 

 this form, is in general an impossible one. 



The modern theory, inaugurated by Briot and Bouquet's and 

 Fuchs's discoveries, has reversed the whole problem. It considers 

 the differential equation (together with a proper number of initial 

 conditions) as defining a function, and proposes to derive directly 

 from the differential equation the characteristic properties of its in- 

 tegrals, true to the general principle of the theory of functions, that 

 the essential thing about a function is not its form, which usually 

 may be varied in many ways, but the totality of its characteristic 

 properties. 



It is in particular the theory of linear differential equations that 

 has been very fully considered from this standpoint ; and there is 

 scarcely any branch of mathematical science that has attracted a 

 more general attention in our day, and in which more important 

 discoveries have been made, than the theory of linear differential 

 equations. Still every one who wished to become familiar with it, 

 and who had to work his way through the vast and difficult litera- 

 ture on the subject, has keenly felt the want of a systematic expo- 

 sition uniting the numerous researches scattered in the different 

 mathematical journals and publications of learned societies. 



To meet this want, and to give an account of the theory as it 

 stands to-day, is the object of the " Treatise on Linear Differential 

 Equations." by Professor Thomas Craig of Johns Hopkins Uni- 

 versity. The first volume, which is to be followed by a second 

 one, is entitled "Equations with Uniform Co-efficients," and deals 

 principally with Fuchs's theory and the investigations immediately 

 connected with it. The rich material has been carefully sifted, and 

 is presented in a clear and intelligible language in the most natural 

 order of ideas. 



An introductory chapter gives the general properties of a system 

 of linear differential equations of a more formal character, among 

 others the well-known theorems on systems of independent partic- 

 ular integrals. 



■ Ne.xt follows an elegant exposition of the theory of linear differ- 

 ential equations with constant co- efficients, where the reader will 

 find, besides Euler's solution, an account of various ingenious 

 methods due to Cauchy, Hermite, and others. 



After these preparations, we are led, in Chapter III., into the 

 very centre of the modern theory ; viz., the determination of the 

 form of the integrals in the region of a critical point. It is first 

 shown, that, if the differential equation be written in the form 

 ^ny d'^—ty 

 1- A \- . . . + pnV = O, 



the critical points of any one of its integrals are always found among 

 the critical points of the system of co- efficients, />i, /j . . . /„. 

 Then Fuchs's theorems concerning the form of the integrals in the 

 region of a critical point are developed with all the details about 

 "groups of integrals " added by Hamburger, Floquet, and others. 



A particular integral is said to be regular in a critical point a, if 

 it remains finite for x=ia after multiplication by some proper power 

 of X — a ; and, in order that all the integrals may be regular in a, it 

 is necessary and sufficient that {x — a)" /« (a=i, 2 ... >i) he 

 holomorphic in a. Chapter IV. contains an account of Frobenius's 

 elegant treatment of this case, and gives a simple criterion for the 

 non-appearance of logarithms. 



The next chapters are devoted to that important class of differ- 

 ential equations (called regular equations) all of whose integrals are 

 regular in all the critical points ; and the fertility of the general 

 methods is abundantly shown in the application to the equation of 

 the second order, in particular that with three critical points, which, 

 on account of its high importance, is very fully treated, with many 

 interesting results concerning Riemann's P-function, spherical 

 harmonics, Bessell's functions, etc. 



The differential equation of the hypergeometric series, to which 

 the above equation can always be reduced, takes such a central 

 place in recent mathematical researches that it well deserves to be 

 considered with all detail, as is done in Chapter VII., which con- 

 tains a reproduction of Goursat's " Thesis on the Hypergeometric 

 Series." 



The theory of irregular integrals is still in a very imperfect state. 

 Chapter IX. gives an account of Frobenius's and Thome's re- 

 searches, and the same subject is treated in Chapter X. by the ele- 

 gant method of decomposition of a differential quantic into sym- 

 bolic prime factors. Interesting special classes of irregular equa- 

 tions will be found in the chapters on Halphen's equations, and ot» 

 equations with doubly periodic co-efficients. 



The two remaining chapters might, it seems to us, as well have- 

 been reserved for the second volume, where the same subjects will 

 be more fully dwelt upon. Still the two conceptions of group and' 

 of invariant of a differential equation which they develop are of so- 

 fundamental importance that they can scarcely be introduced too- 

 soon. 



If the CO, efficients of a linear differential equation are uniform^ 

 functions of .r, any system of n independent particular integrals, 

 submit to a homogeneous linear substitution when the variable- 

 point X describes any closed path in its plane. The entire system 

 of substitutions obtained in this way forms a group, called the- 

 " group of the differential equation." 



The notion of "invariant" of a linear differential equation, on^ 

 the other hand, arises when the given equation is transformed into- 

 another of the same form by the introduction of two new variables.. 

 and its definition is analogous to that of an invariant of an alge- 

 braic quantic. 



We must confine ourselves to these few indications, and refer the-' 

 reader to the book itself for further information. Only then will 

 he obtain an adequate idea of the thoroughness and completeness- 

 with which the subject has been treated. As far as we are able to- 

 judge, no investigation of any importance has been omitted, and 

 the justice and conscientiousness with which continually reference 

 to the original papers is given are a characteristic feature of this- 

 most valuable book, which, we are sure, will contribute a great 

 deal to spread the knowledge of this important discipline. 



We look forward with much interest to the appearance of the- 

 second volume, which will contain, among other things, an exposi- 

 tion of the theory of linear differential equations with algebraic in- 

 tegrals, and of Poincare's theory of Fuchsian groups and Fuchsiarb 

 functions. 



AMONG THE PUBLISHERS. 



The Bulletin of the Ohio Agricultural Experiment Station for 

 October, 1889, is Vol. I, No. i of a technical series, and contains- 

 three articles by Clarence M. Weed, — " Preparatory Stages of the- 

 20-Spotted Ladybird," " Studies in Pond Life," and " A Partial- 

 Bibliography of Insects affecting Clover." 



— The opening article in the December number of Outings 

 " Wabun Anung," by F. Houghton, is a clear description of a tourin^ 

 the region of the Great Lakes. Another article is the "Merits and 

 Defects of the National Guard," by Lieut. W. R. Hamilton. We 

 note further the " Game of Curling," by James Hedley ; "Wheel- 

 ing through the Land of Evangeline ; " " Game Protection ; " " In- 

 stantaneous Photography," by W. I. Lincoln Adams; "Women, 

 and their Guns;" "The Yale Stroke;" "Alligator Shooting in- 

 Florida;" and " Na-ma-go-os," a fishing sketch. 



— John Wiley & Sons have just published " A Hand-Book for 

 Sugar Manufacturers and their Chemists," by Guilford L. Spencer 

 of the United States Department of Agriculture. The volume 

 contains practical instruction in sugar-house control, the diffusion 

 process, selected methods of analysis, reference tables, etc. The 

 essential requirements of a thorough chemical control and superin- 

 tendence of a sugar- factory are carefully described, and only such 

 analytical processes are given as relate to sugar-house products 

 and the waste residues when necessary to a complete control 

 Technical chemical terms have as far as possible been avoided. 

 The little book ought to stimulate our sugar-manufacturers and; 

 their chemists to more extensive investigations and more thorough, 

 work. 



— Messrs. Ginn & Co. announce for publication early in De- 

 cember the first volume of a serial entitled " Harvard Studies in 

 Classical Philology," edited by a committee of the classical instruc- 

 tors of Harvard University. It is the expectation that one volume. 



