ON THE THEORY OF INTEGRAL EQUATIONS. 345 



Report on the History and Present State of the Theory of 

 Integral Equations. By H. Bateman. 



[Ordered by the General Committee to be printed in exten$o.~\ 



Contents 



SECTION. PAGE 



1. Integral Equations of Laplace's Type 345 



2. Fourier's Theorems 348 



3. Applications of Fourier's Formula 349 



4. Abel's Integral Equation 352 



5. Integral Equations with Variable Limits . 355 



6. Physical Applications of the Equations with Variable Limits. Problems of 



Heredity and Hysteresis 351 



7. The Problem of Dirichlet and the Method of Successive Approximations . . 363 



8. Green's Functions 366 



9. Fredholm's Mmhod 369 



10. The Applications of Fredholm's Method to Potential Problems .... 380 



11. Orthogonal and Biorthogonal Systems of Functions 383 



12. Expansions in Series of Orthogonal Functions 385 



13. Expansions in Series of Orthogonal Functions connected with a Linear 



Differential or Integral Equation 386 



14. Integral Equations of the First Kind 388 



15. Canonical Form of an Integral Equation of the First Kind. Definite Functions 390 



16. Multiplication Formula! and their Applications 3j3 



17. Integral Equations in which the Principal Value of the Integral must be 



taken 395 



18. Distributive Operations 396 



19. Connection with the Calculus of Variations 399 



20. Riemann's Problem 402 



21. The Solution of Linear Differential Equations by means of Definite Integrals . 404 



22. Applications to the Partial Differential Equations of Mathematical Physics . 406 



23. Applications to Problems in the Theory of Elasticity 410 



24. Bilinear and Quadratic Forms in an Infinite Number of Variables . . . 4 ! 1 



25. Linear Equations in an Infinite Number of Variables 413 



26. Singular Integral Equations 4 6 



27. Miscellaneous Physical Applications 420 



28. Non-linear Integral Equations 423 



1. Integral Equations of Lapilace's Type. 



The theory of integral equations may be said to have commenced in 

 1782, when Laplace ' used definite integrals of the type 



[e^{t)dt =f(x) .... (1) 



to solve linear difference and differential equations, for in these 

 investigations he gave a method of determining the unknown function 

 <p(t), appearing under the sign of integration, when a linear differential 

 equation satisfied by the function f(x) is known. The determination 

 of (f>(t) depends upon the solution of a corresponding linear differential 



1 ' Memoire sur les approximations des formules qui sont fonctions de tres grands 

 nombres,' CEuvres, t. 10, p. 235; Theorie analytique des probability (Paris: 1812). 



