MATHEMATICAL PHYSICS. 387 



2. Definitions of limit points. Existence of a limit point in an infinite set of points. 



3. Types of continuity and relations among them. 



4. Convergent functions and uniformly convergent functions. 



5. Independence of continuity and convergence. 



6. Representation of convergent functions by series. 



7. A continuous function of infinitely many variables in a closed P-space has a maxi- 



mum and a minimum and all intermediate values. 



8. Definition of analytic functions of infinitely many variables. 



9. Definition of normal functions of infinitely many variables. 



10. Finite operations on functions of infinitely many variables. 



11. Limiting processes on functions of infinitely many variables. 



12. The mean-value theorem for completely continuous functions having first deriva- 



tives. 



13. Taylor's theorem for functions of infinitel}' many variables. 



III. Infinite Systems of Implicit Function Equations. 



1. Analytic solutions of reduced normal equations. 



2. Analytic continuation of the solutions. 



3. Solutions of normal equations ha\'ing normal determinants of the coefficients of the 



linear terms of the dependent variables. 



4. Properties of the solutions of normal equations. 



5. Solution of reduced normal equations by the method of successive approximations. 



6. Extension of the solution to a boundary of the region of definition of the equations. 



7. Solutions of infinite systems of equations having continuous first derivatives. 



8. Properties of the solutions. 



9. Extension of the solution to a boundary of the region of definition of the equations, 



IV. Infinite Systems of Differential Equations. 



1. Analytic solutions of normal equations. 



2. Solution of normal equations by the method of successive approximations. 



3. Solution of equations satisfying the Lipschitz condition by the method of successive 



approximations. 



4. Properties of the solutions. 



5. Extension of the solution to a boundary of the region for which the equations are 



defined. 



6. Solution by the Cauchy-Lipschitz method. 



7. Solutions of infinite systems of hnear differential equations having constant coeffi- 



cients. 



8. Solutions of infinite systems of linear differential equations having periodic coeffi- 



cients. 



V. Applications of Functions of Infinitely Many Variables. 



1. Hill's problem of the motion of the lunar perigee. 



2. Solutions of linear differential equations in the vicinity of singular points. 



3. The determination of the moon's variational orbit. 



4. Determination of periodic solutions of certain finite systems of differential equations. 



5. The dynamics of a certain type of infinite universe. 



A chapter on Numerical Integration of Differential Equations has 

 been wTitten for a volume which is being prepared for the Smithsonian 

 Institution, by Professor E. P. Adams. 



