Kessinger Publishing, LLC, 2007. — 360 pages.This book is a facsimile reprint and may contain imperfections such as marks, notations, marginalia and flawed pages.Initial explanations
Some typical forms of equations to be considered
Weierstrass's normal form for a system
Preparation of normal forms for consideration
Note. Weierstrass's theorem on the form of a regular function of several variables in the vicinity of a zero valueCauchy's fundamental theorem on the regular integrals of a system of differential equations stated, and proved.
Uniqueness of the regular system
Jordan's proof of the uniqueness
Bibliographical NoteContinuation of regular integrals: exceptional or critical points
On the possibility of non-regular integrals, satisfying the same conditions as regular integrals
Classes of exceptional points indicated by the form of the differential equation
Various classes of singularities of the equation when it is uniform: with respective typical expressions valid in their vicinity
Discrimination between isolated singularities and continuous singularities
Branch-points, as determined by the form of the equationCharacter of the integral in the vicinity of an accidental singularity of the first kind: first case, when the point is an algebraical branch-point of the integral .
Second case, when the integral does not exist: or, if it exists, has the point for an essential singularity
Infinite values of the integral function
Some examples
Non-regular integrals : Fuchs's points of indeterminateness
Fuchs's methods of discussing the character of the integral in the vicinity of possible points of indeterminateness: with examples
General summary of results, relating to the vicinity of an accidental singularity of the first kind
On the uniqueness of Cauchy's regular integral: Fuchs's condition, with an exampleComprehensive form of the equation in the immediate vicinity of an accidental singularity of the second kind
Preliminary discussion of simplest case
Puiseux diagram applied to obtain the leading term of an integral in the vicinity of a singularity; with an example, and remarks
Construction of the various typical reduced forms of equation in the first and most general instance
Construction of these forms in the remaining instancesTransformation of the first typical reduced form Case I, when a is not a positive integer; Briot and Bouquet's theorem as to regular integrals .
Theorems as to non-regular integrals in Case I, due to Briot and Bouquet, and to Poincare; Jordan's proof
Summary of results in Case I
Case II, when a is a positive integer 0: indications derived by making this a limit of the first Case
riot and Bouquet's theorem on the regular integrals of the equation, when a is unity
Poincare's theorem on the non-regular integrals of the equation, when a is unity , Jordan's proof of Poincare's theorem
Explicit expression of the non-regular integral, when a is a positive integer: with two Corollaries
The non-existence of non-regular integrals in general
Examples of this form
Two omitted cases, with examples
The remaining typical forms, with examples
General summary of results relating to all the typical forms
The second typical form t -j-.=gV
