BOOK I. GENERAL PROPERTIES OF CAUCHY'S PROBLEM

  I. CAUCHY'S FUNDAMENTAL THEOREM CHARACTERISTICS

  II. DISCUSSION OF CAUCHY'S RESULT

BOOK II. THE FUNDAMENTAL FORMULA AND THE ELEMENTARY SOLUTION

  I. CLASSIC CASES AND RESULTS

  II. THE FUNDAMENTAL FORMULA

  III. THE ELEMENTARY SOLUTION

    1. GENERAL REMARKS

    2. SOLUTIONS WITH AN ALGEBROID SINGULARITY

    3. THE CASE OF THE CHARACTERISTIC CONOID.

      THE ELEMENTARY SOLUTION

    ADDITIONAL NOTE ON THE EQUATIONS OF GEODESICS

BOOK III. THE EQUATIONS WITH AN ODD NUMBER OF INDEPENDENT VARIABLES

  I. INTRODUCTION OF A NEW KIND OF IMPROPER INTEGRAL

    1. DISCUSSION OF PRECEDING RESULTS

    2. THE FINITE PART OF AN INFINITE SIMPLE INTEGRAL

    3. THE CASE OF MULTIPLE INTEGRALS

    4. SOME IMPORTANT EXAMPLES

  II. THE INTEGRATION FOR AN ODD NUMBER OF INDEPENDENT VARIABLES

  III. SYNTHESIS OF THE SOLUTION OBTAINED

  IV. APPLICATIONS TO FAMILIAR EQUATIONS

BOOK IV. THE EQUATIONS WITH AN EVEN

    NUMBER OF INDEPENDENT VARIABLES AND THE METHOD OF DESCENT

  I. INTEGRATION OF THE EQUATION IN 2ml VARIABLES

    1. GENERAL FORMULAE

    2. FAMILIAR EXAMPLES

    3. APPLICATION TO A DISCUSSION OF CAUCHY'S PROBLEM

  II. OTHER APPLICATIONS OF THE PRINCIPLE OF DESCENT

    1. DESCENT FROM zn EVEN TO m ODD

    2. PROPERTIES OF THE COEFFICIENTS IN THE ELEMENTARY SOLUTION

    3. TREATMENT OF NON-ANALYTIC EQUATIONS

INDEX