ESSAYS 



On the Relation of the Theory of Integral Equations to the Subject of 

 the Calculus of Operations and Functions (H. Bateman, M.A., Sc.D., 

 Lecturer at John Hopkins University, Baltimore, U.S.A.) 



It has been pointed out by Pincherle ' that many of the theorems of a general 

 nature in the theory of integral equations are simply illustrations of theorems 

 belonging to the general theory of distributive operations, and this is one of the 

 leading ideas in the development of E. H. Moore's " General Analysis." 2 

 Moore puts the matter in this way : " The existence of analogies between central 

 features of various theories implies the existence of a general theory which 

 underlies the particular theories and unifies them with respect to those central 

 features." In other words, in the general theory of functional equations there are 

 certain fundamental ideas of supreme importance for the development of all the 

 branches of the subject. What are these fundamental ideas ? This is the question 

 we shall now endeavour to answer. 



In the first place there is the idea of iteration, or the repetition of a single 

 mathematical operation such as multiplication by a certain fixed number. This 

 idea was used by Michael Dary 3 in his method of solving equations (which was 

 described in Science Progress, October 191 5, and January and April, 1916). 



Let x — F(x) be the equation to be solved and let x be a number which 

 is somewhere near the root. If we draw the curves y = F(x), y = x, we can 

 frequently approach a point of intersection by a series of steps represented 

 analytically by the series of equations. 



Xx = Fixu), x = F(x{), x s = F(x.,\ , 



x = lim x 

 In this case the operation which is repeated is that of forming a given function 

 of the quantity x. In the theory of integral equations the operation which is 

 repeated is of the following type : 



/!(,) =f b k(s,t)flt)dt, f 2 {s) =f b k(s,t)flt)dl, fls) =f'je{s,t)f.,(t)df, 



and if we introduce a parameter X, Dary's equation 



x = \F(x) 

 has as its analogue the homogeneous integral equation of the second kind. 

 As) = \fjc(s,tW)dt. 



1 Rend. Lincei, 1905 ; Bologna Memoirs (6), 1906, 3 ; 191 1, 8. 



2 Introduction to a Form oj General Analysis, New Haven Mathematical 

 Colloquium (1906), New Haven, 1910; Borne Mathematical Congress (1908), 

 Atti, 1909, 2, 98 ; Bulletin of the American Mathematical Society, April 1912 ; 

 Cambridge Math. Congress (191 2), 1, 230. 



3 August 15, 1674. An account of Dary's work is given by W. Stott, SCIENCE 

 Progress, October 191 5. 



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