Colloquium publications.

CONTENTS

xi

LECTURE IV
INTEGRO-DIFFERENTIAL EQUATIONS OF BOCHER TYPE
~ 1. The Generalization of Laplace's Equation. Arts. 46-48. 73
Hypothetical experiments as a basis of Physics.
Bocher's treatment of Laplace's equation. Poisson's equation.
~ 2. Green's Theorem for the General Linear Integro-Differential Equation of Bocher Type. Arts. 49-53....... 78
Adjoint integro-differential equations. Proof of
Green's theorem. A proof by approximating polynomials. Change of variable. The three types of
equation.
~ 3. The Parabolic Integro-Differential Equation of Bocher
Type. Arts. 54-58............................. 92
Derivation of the equation for the flow of heat.
The Dirichlet problem. The uniqueness of the
solution. Existence of solutions. The Green's
function.
~ 4. The Parabolic Integro-Differential Equation of the Usual
Type. A rt. 59................................. 98
The generalized Green's function.
~ 5. The Differential Equation of Hyperbolic Type. Art. 60. 99
Functions of zero variation in two dimensions.
LECTURE V
DIRECT GENERALIZATIONS OF THE THEORY OF INTEGRAL
EQUATIONS
~ 1. Introduction: Some General Properties of the Stieltjes
Integral. Arts. 61-62........................... 101
The Stieltjes integral equation. Integrability of
the Stieltjes integral.
~ 2. The General Analysis of E. H. Moore. Arts. 63-70... 105
Possible points of view. The linear equation and
its kernel. Bases and postulates for equation (G).