A major portion of the study of the qualitative nature of solutions of differential equations may be traced to the famous 1836 paper of Sturm [1), (here, as elsewhere throughout this manuscript, numbers in square brackets refer to the bibliography at the end of this volume), dealing with oscilla- tion and comparison theorems for linear homogeneous second order ordinary differential equations. The associated work of Liouville introduced a type of boundary problem known as a "Sturm-Liouville problem", involving, in particular, an introduction to the study of the asymptotic behavior of solu- tions of linear second order differential equations by the use of integral equations. In the quarter century following the 1891 Gottingen dissertation [1) of Maxime Bacher (1867-1918), he was instru- mental in the elaboration and extension of the oscillation, separation, and comparison theorems of Sturm, both in his many papers on the subject and his lectures at the Sorbonne in 1913-1914, which were subsequently published as his famous Leaons sur Zes methodes de Sturm [7).

A major portion of the study of the qualitative nature of solutions of differential equations may be traced to the famous 1836 paper of Sturm [1), (here, as elsewhere throughout this manuscript, numbers in square brackets refer to the bibliography at the end of this volume), dealing with oscilla- tion and comparison theorems for linear homogeneous second order ordinary differential equations. The associated work of Liouville introduced a type of boundary problem known as a "Sturm-Liouville problem", involving, in particular, an introduction to the study of the asymptotic behavior of solu- tions of linear second order differential equations by the use of integral equations. In the quarter century following the 1891 Gottingen dissertation [1) of Maxime Bacher (1867-1918), he was instru- mental in the elaboration and extension of the oscillation, separation, and comparison theorems of Sturm, both in his many papers on the subject and his lectures at the Sorbonne in 1913-1914, which were subsequently published as his famous Leaons sur Zes methodes de Sturm [7).