364 REPORTS ON THE STATE OF SCIENCE. 



the values of the normal derivatives -_-? , -__ * just outside and just in- 



on on J 



side the boundary are connected by the equations 



G& + isw *«**«* 



Tbe potential of a double layer is discontinuous at the boundary, its 

 values, W,, W , just inside and just outside being connected by the 

 equations ' 



l[Wi-WJ «,»(«) 



hlW i + \N l ] = L(*)h(„,s)d<T. 



Poincare enunciates the two problems of potential theory in the 

 following form — 



I. To find a potential of a double layer which satisfies the condition 



1[W«-wj-W + wj=/w 



at a regular point t of the boundary.-' 



II. To find a potential of a simple layer which satisfies the condition 



r8v _av,i _ x rtv, ay, i _ 



\_dn dn] 2 \_dn + dn J ~ /W> 



These problems are reduced at once to the solution of the adjoint 

 integral equations 



MO - 4n(s)h(s,t)ds = f(t) 

 ,,(t)-\ih(t,s)t,(s)ds=f(t). 



For A = — 1 we have the internal problem of Dirichlet and the 

 external hydrodynamical problem ; for \ = + 1 we have the external 

 problem of Dirichlet and the internal hydrodynamical problem. Poincare 

 calls the general problem in which the parameter X occurs the problem 

 of Neumann. The introduction of the parameter X by Poincare is really 

 a very important step. 



In Neumann's method of solution the functions i*{t), p{t) are deter- 

 mined by expanding them in powers of X and determining the coeffi- 

 cients in succession. This method, which has been applied to differential 

 equations since the time of Laplace, 2 has been called by Picard the 

 method of successive approximations. The method has been greatly 



1 These relations are of course classical. Careful proofs of them are given by 

 Plemelj, Monatshefte ficr Math. u. Physik, 1904, and in a little book by "Viscount 

 cl'Adhemar, Sur V equation de Fredholm et les probUmet de Dirichlet fit dc Neumann, 

 (Paris) 1909. » (Eurres, t. 10, p. 54, 1779. 



