The present monograph has points in common with two branches of analysis. One of them is the variational-difference method (the finite element method), the other is the constructive theory of functions. The starting point is the construction of special classes of coordinate functions for the variational-difference method. It is based on elementary transformations .of the independent variables of given "primitive" functions. After the construction of the coordinate functions, the next step is to approximate functions of a given class by linear combinations of the coordinate functions, and to derive in some appropriate norm an estimate of the error. Clearly, this is a problem closely connected with the constructive theory of functions. The monograph contains 11 chapters. Chapter I discusses Courant's basic idea which is central to the construction of variational-difference methods. One of Courant's examples, from which the notion of a primitive function follows naturally, is examined in some detail. The general definition of a primitive function and the method of construction for the corresponding coordinate functions are given and discussed. Chapters II-VI are more closely connected with the constructive theory of functions. The completeness of the coordinate systems defined in Chapter I are studied, as well as the order of approximation obtained through the use of linear combinations of these functions. Their completeness in Sobolev spaces are examined in Chapter II, while related orders of approximation are derived in Chapter III.

It was the last book the outstanding mathematician, mechanician and lecturer S. G. Mikhlin took an active part in writing. Having been completed during his lifetime, this book could not be published in Russia due to well- know difficulties. Since that time new results in integral equations of elasticity theory have appeared. The works of W. Wendland and his school on numerical methods of solving boundary integral equations, the works of I. Chudinovich on inves- tigation of non-stationary integral equations, the works of S. Kuznetsov con- nected with the construction of the fundamental solutions for anisotropic me- dia and others deserve special mentioning. The authors recognize that though the book is devoted to integral equations of elasticity theory, its contents do not cover all possible directions in this field. So the book does not contain the investigations of pseudo-differential equations of three-dimensional prob- lems of elasticity theory, connected with the works of R. Goldstein, I. Klein, G. Eskin; the questions of solving by integral transformations (I. Ufland, L. Slepian, B. Buda. e: v); the theory of symbols of pseudo-differential operators on non-smooth surfaces developed in the works of B. Plamenevski et al. and the new methods of numerical solution of pseudo-differential equations as developed by a school of V. Mazya. The present book gives the classical methods of potential theory in elas- ticity and their development and also the solution of a number of problems which here are published in English for the first time.