382 REPORTS ON THE STATE OF SCIENCE. 



satisfies the equation 



W) = \\g{s,r)h( S ,t),p(t)dsdt 



- \[h{s,r)g{ S) i)f{t)dsdt 

 on account of the symmetry of K(r,t) 



and so H T )= \h{s,r)xL(s)ds. 



Hence since it can be shown that X = 1 is only a simple root for the 

 integral equation the function \p(s) must be a constant, and so the 

 potential of a simple layer having <{>(s) as density is constant over the 

 boundary. 



Lauricella and Picard have also considered the problem of deter- 

 mining a potential of a double layer which shall coincide with a 

 potential of a simple layer in one of the regions of space. The problem 

 depends on a double application of the preceding problems and provides 

 a method of solving a type of integral equation of the first kind. 



Plemelj has shown further how one can proceed to determine the 

 Green's function, and he introduces a more general type of Green's 

 function and discusses its properties. 



Plemelj 's discussion of the potential problems is not confined to a 

 single closed surface ; he considers a boundary consisting of a number 

 of closed surfaces, some of which surround a number of others. In the 

 general case both \ = — 1 and A = + 1 are roots of the kernel h(s,t), 

 and are in fact multiple roots. It follows however from the symmetry 

 of k(s,t) that the solving function has only simple poles which are all 

 real. The theory is considerably simplified by this circumstance. The 

 solutions of the homogeneous equation analogous to (2) provide the 

 densities of potential functions which are constant over one surface and 

 zero over all the others. 



Plemelj has also applied the theory to more general electrostatic 

 problems in which surfaces of discontinuity of the potential are con- 

 sidered. 



The modifications introduced into the discussion of the problems 

 when the boundary has corners or conical points are considered by 

 Kellogg. He shows that in the case of a curve with a sharp angle w at 

 s = 0, 



, ,. . sin w s j_ tp /<• a s < 



)l(t,s) = a 5-: —7, + &l(S,t) f ^ n 



v ' tt s- — 2st cos w 4- r t > u 



sin w s j v fc a s > 



= ~T «' - 2s* cos u, + *'-' + ^ ,t] t < 



where E,(s,i) E a (s,£) are finite for s = t = 0. The case of a cusp is 

 exceptional. 



For such a kernel h(t,s) the series for D(X ; s,t) and D(A) in Fred- 

 holm's theory are meaningless since h(s,t) becomes infinite to too 

 high an order. This is certainly the case, for instance, when w is a right 

 angle. 



