3GG REPORTS ON THE STATE OF SCIENCE. 



and the solving function 



K(s,0 = u(s,t) + \K-,(s,t) + \^s(s,t) + 



b 



This series has a finite radius of convergence if [^{x,y)fdy and 



V 



K^.2/)] 2 ^ are always finite, 1 or if I u(x,y)dxdy is convergent. If \ 



a a a 



has a value whose modulus is less than the radius of convergence and 

 f(s) is bounded, the solution of the integral equation is given uniquely 

 by the equation 



b 



<t>(s)=f( S ) + ^K( S) t)f(t)dt, 



and the relation between the functions ^(s,t), K(s,t) is of a reciprocal 

 nature. These formulae are established for other values of X by 

 Fredholm's method. 



It is, in fact, a consequence of Poincare's researches that the 

 functions (p(s), K(s,i) are mcromorphic functions of \, and Fredholm states 

 at the beginning of his paper that his method provides a direct method 

 of obtaining these functions in the canonical form of the ratio of two 

 integral functions of A. 



8. Greens Functions. 



The function which Neumann has called the Green's function was first 

 introduced by George Green in his essay on the application of mathe- 

 matical analysis to the theories of electricity and magnetism (1828), 

 p. 26. 



The Green's function G {x,y,s; £,»/,4') is defined to be a function 

 which satisfies Laplace's equation v • V = 0, and is a regular analytic 

 function of x,y,z, at all points inside (or outside) the surface S with the 

 exception of the point (£,»?,£). In the neighbourhood of this point 



G i^_ . . m 



V{x - If + (y - v) 2 + (* - t) 2 K ] 



s a regular analytic function. On the boundary of S, G satisfies a 



8G 8G 

 linear condition such as G = 0, ~- =0, or ,--: 1- hG = 0. The funda- 



on On 



mental property of the Green's function is that a solution of v 2 V 

 + f(x,y,z) = which satisfies the same boundary condition as G, is 

 given by the formula 



V(.r,?/,2) = JljG^gr,*; ?,ij,£) f f faf,Qd&kAC. 



This equation may be regarded as an integral equation for the deter- 

 mination of / when V is given. If V satisfies the boundary condition 



1 E. Schmidt, Math . Aim. 



