## Linear Differential Equations by Thomas Craig

Thomas Craig published A Treatise on Linear Differential Equations in 1889. We give below an extract from Craig's Preface to this work as well as an extract from a review.

See THIS LINK for extracts from the Prefaces of two other books by Craig, namely Wave and Vortex Motion (1879), and A Treatise on Projections (1880).

1. Preface to A Treatise on Linear Differential Equations.

The theory of linear differential equations may almost be said to find its origin in Fuchs's two memoirs published in 1866 and 1868 in volumes 66 and 69 of Crelle's Journal. Previous to this the only class of linear differential equations for which a general method of integration was known was the class of equations with constant coefficients, including of course Legendre's well-known equation which is immediately transformable into one with constant coefficients. After the appearance of Fuchs's second memoir many mathematicians, particularly in France and Germany, including Fuchs himself, took up the subject which, though still in its infancy, now possesses a very large literature.

This literature, however, is so scattered among the different mathematical journals and publications of learned societies that it is extremely difficult for students to read up the subject properly.

I have endeavoured in the present treatise to give a by no means complete but, I trust, a sufficient account of the theory as it stands today, to meet the needs of students. Full references to original sources are given in every case.

Most of the results in the first two chapters, which deal with the general properties of linear differential equations and with equations having constant coefficients, are of course, old, but the presentation of these properties is comparatively new and is due to such mathematicians as Hermite, Jordan, Darboux, and others. All that follows these two chapters is quite new and constitutes the essential part of the modern theory of linear differential equations.

The present volume deals principally with Fuchs's type of equations, i.e. equations whose integrals are all regular; a sufficient account has been given, however, of the researches of Frobenius and Thomé on equations whose integrals are not all regular. A pretty full account, due to Jordan, has been given of the application of the theory of substitutions to linear differential equations. This subject will, however, be very much more fully dwelt upon in Volume II, where I intend to take up the question of equations having algebraic integrals and also give an account of Poincaré's splendid investigations of Fuchsian groups and Fuchsian functions. The theory of invariants of linear differential equations has been several times touched upon in the present volume, and some of the simpler results of the theory have been employed; but its extended development is necessarily reserved for Volume II, as is also the development of Forsyth's associate equations, about which extremely interesting subject very little is as yet known.

The equation of the second order with critical points 0, 1, infinity, has on account of its great importance been very fully treated. In connection with this subject it seemed to me that I could not possibly do better than reproduce ... Goursat's Thesis on equations of the second order satisfied by the hypergeometric series. M Goursat was kind enough to give me permission to make a translation of his Thesis, which is, I imagine, not very well known among English and American students.
...

I wish to tender my thanks to M Goursat for his kindness in permitting me to make a translation of his most valuable Thesis, and to Dr Oskar Bolza, Mr C H Chapman, and Dr J C Fields for much valuable assistance.

Thomas Craig
Johns Hopkins University
Baltimore, 1889.

2. Review by: Anon.
Science 14 (357) (1889), 391-392.
The theory of differential equations has undergone within the last thirty years a most fundamental change. The object of the older theory was to integrate a given differential equation "in finite 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 integrals, 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 literature on the subject, has keenly felt the want of a systematic exposition 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 today, is the object of the Treatise on Linear Differential Equations, by Professor Thomas Craig of Johns Hopkins University. The first volume, which is to be followed by a second one, is entitled Equations with Uniform Coefficients, 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. ...

We ... 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 exposition of the theory of linear differential equations with algebraic integrals, and of Poincaré's theory of Fuchsian groups and Fuchsian functions.

JOC/EFR October 2015