380 REPORTS ON THE STATE OF SCIENCE. 

 1 



can be used as an iterated kernel is bounded. 



h 



Additional Results. 



Volterra's integral equation may be deduced from that of Fredholm 

 by writing 



,,(s,t) = (j(s,l) t<s 



= t > s. (Fredholm) 



In the case of Hilbert's kernel 



4M) = s(l - t) <t 



t(l — s) t < s 



the roots are ir 2 , (2t) 2 , (3t) 2 , and the normal functions 



V2 sin (mrs). 



If r(x) is continuous in the interval (a,b) and does not vanish, the 

 solving function for the kernel 



. , r(x) , 

 9( x ' t ) = r(t) K ( x,i > 



1S G(x,t) = -f^ K(x,t). , r ,. 



K ' r (t) (Goursat) 



An integral equation with a kernel of form r(x)g(x,t) where g(x,t) is 

 symmetric can be reduced to the symmetric form by multiplying 

 throughout by 



./g{t) 



V ij(x) (Schmidt and Goursat) 



If K(s,t) = — ^{t,s), the roots are all purely imaginary quantities, but 

 there are at least two. 



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



The theory of the boundary problems of potential theory from the 

 point of view of Fredholm's work has been worked out very fully by 

 Fredholm, Hilbert, Kellogg, 1 Andrae, 2 Plemelj, 3 Picard, 4 and Lauricella. 5 

 Plemelj's papers contain almost an exhaustive discussion of the 

 problems. We have seen that the theory depends upon the properties 

 of the two adjoint integral equations 



f(t) = p(t) — \L{s)h(s,t)ds 

 f(t) = P {t) - \{h(t,s) P (s)ds, 



1 Dissertation Gottingen, 1902 ; Math. Ann., 1903, Bd. lviii.; 1904 ; Bd. lx. 



* Dissertation Gottingen, 1903. 



3 Monatsheftefur Math, in Phyt., 1904 and 1906. 



* Rend. Palermo, 1906. See also Compf.es rendus, 1905-08. 

 5 11 Niwvo Cimento, 1907, ser. 6, vol. xiii. 



