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VI. 



ON AN INTEGRAL EQUATION PROPOSED BY ABEL, AND 

 OTHER FUNCTIONAL EQUATIONS RELATED TO IT.* 



By REV. PATRICK J. BROWNE, M.A., D. es Sc, 

 Professor of Mathematics, St. Patrick's College, Maynooth. 



Read April 12. Published September 25, 1915. 



1. The integral equation, in the form in which Abel proposed it, is as 



follows : — 



jf(x) (p (ax) dx = \p (a), 



where / and \p are known, and </> is to be found. The limits of integration 

 are constants. Abel gave the equation as a generalization of the problem 

 of the isochrone, stating that he had solved it, but did not give the solution. 

 We will first take the limits of integration to be and 1, and write the 

 equation thus — 



' X G(t)f(tx)dt = g{x). 



Here G and g are known, while / is unknown. This becomes, on inte- 

 gration by parts, 



f 1 

 G{l)$(x)- G'(t) <j>(tx)dt = xg(x), 



Jo 



where <j> (x) = f(x) dx. 



. 



Supposing G (1) =(= 0, we may write 



•i 

 <\> (x) = i/< (*) + K(t) <p (tx) dt.f 

 \ Jo 



* See the author's These de Boctorat (Paris, 1913), since published in the Annales de Toulouse 

 (1914). A brief history of the question is there given, and a proof of the existence of solutions. 



t The substitution tx - y brings this equation to Volterra's form with a kernel 



1 



©• 



which is not limited in the neighbourhood we are considering, i.e. that of x = 0. Still, the equation 

 can be solved by iteration when \|/ (x) is limited and | K(t)\ < 1. 



E.I.A. PROC, VOL. XXXII., SECT. A. [9] 



