Chapter I. The Force of Gravity. |

1. The Subject Matter of Potential Theory |

2. Newton's Law |

3. Interpretation of Newton's Law for Continuously Distributed Bodies |

4. Forces Due to Special Bodies |

5. "Material Curves, or Wires" |

6. Material Surfaces or Laminas |

7. Curved Laminas |

8. "Ordinary Bodies, or Volume Distributions" |

9. The Force at Points of the Attracting Masses |

10. Legitimacy of the Amplified Statement of Newton's Law; Attraction between Bodies |

11. Presence of the Couple; Centrobaric Bodies; Specific Force |

Chapter II. Fields of Force. |

1. Fields of Force and Other Vector Fields |

2. Lines of Force |

3. Velocity fields |

4. "Expansion, or Divergence of a Field" |

5. The Divergence Theorem |

6. Flux of Force; Solenoidal Fields |

7. Gauss' Integral |

8. Sources and Sinks |

9. General Flows of Fluids; Equation of Continuity |

Chapter III. The Potential. |

1. Work and Potential Energy |

2. Equipotential Surfaces |

3. Potentials of Special Distributions |

4. The Potential of a Homogenous Circumference |

5. Two Dimensional Problems; The Logarithmic Potential |

6. Magnetic Particles |

7. "Magnetic Shells, or Double Distributions" |

8. Irrotational Flow |

9. Stokes' Theorem |

10. Flow of Heat |

11. The Energy of Distributions |

12. Reciprocity; Gauss' Theorem of the Arithmetic Mean |

Chapter IV. The Divergence Theorem. |

1. Purpose of the Chapter |

2. The Divergence Theorem for Normal Regions |

3. First Extension Principle |

4. Stokes' Theorem |

5. Sets of Points |

6. The Heine-Borel Theorem |

7. Functions of One Variable; Regular Curves |

8. Functions of Two Variables; Regular Surfaces |

9. Function of Three Variables |

10. Second Extension Principle; The Divergence Theorem for Regular Regions |

11. Lightening of the Requirements with Respect to the Field |

12. Stokes' Theorem for Regular Surfaces |

Chapter V. Properties of Newtonian Potentials at Points of Free Space. |

1. Derivatives; Laplace's Equation |

2. Development of Potentials in Series |

3. Legendre Polynomials |

4. Analytic Character of Newtonian Potentials |

5. Spherical Harmonics |

6. Development in Series of Spherical Harmonics |

7. Development Valid at Great Distance |

8. Behavior of Newtonian Potentials at Great Distances |

Chapter VI. Properties of Newtonian Potentials at Points Occupied by Masses. |

1. Character of the Problem |

2. Lemmas on Improper Integrals |

3. The Potentials of Volume Distributions |

4. Lemmas on Sur |

5. The Potentials of Surface Distributions |

6. The Potentials of Double Distributions |

7. The Discontinuities of Logarithmic Potentials |

Chapter VII. Potentials as Solutions of Laplace's Equation; Electrostatics. |

1. Electrostatics in Homogeneous Media |

2. The Electrostatic Problem for a Spherical Conductor |

3. General Coördinates |

4. Ellipsoidal Coördinates |

5. The Conductor Problem for the Ellipsoid |

6. The Potential of the Solid Homogeneous Ellipsoid |

7. Remarks on the Analytic Continuation of Potentials |

8. Further Examples Leading to Solutions of Laplace's Equations |

9. Electrostatics; Non-homogeneous Media |

Chapter VIII. Harmonic Functions. |

1. Theorems of Uniqueness |

2. Relations on the Boundary between Pairs of Harmonic Functions |

3. Infinite Regions |

4. Any Harmonic Function is a Newtonian Potential |

5. Uniqueness of Distributins Producing a Potential |

6. Further Consequences of Green's Third Identity |

7. The Converse of Gauss' Theorem |

Chapter IX. Electric Images; Green's Function. |

1. Electric Images |

2. Inversion; Kelvin Tranformations |

3. Green's Function |

4. Poisson's Integral; Existence Theorem for the Sphere |

5. Other Existence Theorems |

Chapter X. Sequences of Harmonic Functions. |

1. Harnack's First Theorem on Convergence |

2. Expansions in Spherical Harmonics |

3. Series of Zonal Harmonics |

4. Convergence on the Surface of the Sphere |

5. The Continuation of Harmonic Functions |

6. Harnack's Inequality and Second Convergence Theorem |

7. Further Convergence Theorems |

8. Isolated Singularities of Harmonic Functions |

9. Equipotential Surfaces |

Chapter XI. Fundamental Existence Theorems. |

1. Historical Introduction |

2. Formulation of the Dirichlet and Neumann Problems in Terms of Integral Equations |

3. Solution of Integral Equations for Small Values of the Parameter |

4. The Resolvent |

5. The Quotient Form for the Resolvent |

6. Linear Dependence; Orthogonal and Biorthogonal Sets of Functions |

7. The Homogeneous Integral Equations |

8. The Non-homogeneous Integral Equation; Summary of Results for Continuous Kernels |

9. Preliminary Study of the Kernel of Potential Theory |

10. The Integral Equation with Discontinuous Kernel |

11. The Characteristic Numbers of the Special Kernel |

12. Solution of the Boundary Value Problems |

13. Further Consideration of the Dirichlet Problem; Superharmonic and Subharmonic Functions |

14. Approximation to a Given Domain by the Domains of a Nested Sequence |

15. The Construction of a Sequence Defining the Solution of the Dirichlet Problem |

16. Extensions; Further Propeties of U |

17. Bar |

18. The Construction of Barriers |

19. Capacity |

20. Exceptional Points |

Chapter XII. The Logarithmic Potential. |

1. The Relation of Logarithmic to Newtonian Potentials |

2. Analytic Functions of a Complex Variable |

3. The Cauchy-Riemann Differential Equations |

4. Geometric Significance of the Existence of the Derivative |

5. Cauchy's Integral Theorem |

6. Cauchy's Integral |

7. The Continuation of Analytic Function |

8. Developments in Fourier Series |

9. The Convergence of Fourier Series |

10. Conformal Mapping |

11. Green's Function for Regions of the Plane |

12. Green's Function and Conformal Mapping |

13. The Mapping of Polygons |

Bibliographical Notes |

Index |