769 



I take the opportunity to correct an error in ray former communi- 

 cation (quoted abovel. On page 1003 in tlie equation that follows 

 equation (11) and in the next equation 3 r^ P instead of ?'■' (2P -|- Q) 

 is to be read. This makes (12) valid in any case (and not only in 

 the case of a liquid), even when (S=|=0 /- being <[ /?. (13) of course 

 is valid only, as before, in the case of S being zero outside a 

 sphere of radius R at points in whicli r ^ R. 



Mathematics. — "ö?z a linear integral equation of Volterra of 



Ike first kind, loliose kernel contains a function of Bkssel." 

 By Dr. J. G. Rutgers. (Communicated by Prof. W. Kapteyn). 



(Qoinmunicated in the meeting of September 25, 1915). 



Among the few applications given of Sonine's extension of Abel's 

 integral equation ') we may arrange the integral equation : 



/(.f)=r(i-^)r|JJ(x-gr^i_,(.V^=|)«(§)d5. . (1) 



a 



with the solution: 



^ 1—;. 



ill which it is supposed: 0<:^A<[1, and the given function ƒ(.«) 

 must satisfy the conditions that f{x) is an analytical function, ƒ'(«) 

 finite with at most a finite number of discontinuities for a^x^b, 

 and f {a) = 0. 



In what follows we shall prove that i^l) and (2) -may be directly 

 deduced by means of relations known in the theory of the functions 

 of Bessel. In this deduction it will moreover appear that the solution 

 of (1) only becomes the form (2) if a definite value is avoided and 

 consequently a great restriction is imposed on a certain parameter, 

 what is not strictly necessary. This special selection has moreover 

 the drawback that we only come to the solution of (1) under the 

 very restrictive condition <^ ii (P.) <[ 1 , whereas the more general 

 expression which we may find in this way, gives a solution under 

 the far more extensive condition R(X)<:^1. The conditions which 

 the given function f{x) must satisfy will manifest themselves. 



1) SoNiNE : Acta Matem. 4 (1884). These Proceedings XVI p. 583 (1913). 



