288 -42- 



and has the following boundary conditions: 



- •=—2. = A' - B' cos on the sphere 

 3 n 



(1) [ 



d <i 



- ■^-5. = on the plane, 



where n is the normal direction pointing into the liquid 

 and the "prime" denotes a time derivative. 



The construction of $ can be decomposed into two 

 simpler problems by setting 



(2) 4> = A« <|>Q - B' <^^ 



where <^ , <|> -, are potential fvuictions satisfying the 

 following boundary conditions: 



(3) - 1^ = 1 on the sphere, - ^-^ = 



on the plane 



3 <5 1 3^-1 



(4) - i = cos © on the sphere, ^-^-1 = on the plane. 



Physically, <|> represents the potential function for a 

 sphere of fixed center and radius expanding with unit velocity, 

 while ^, is the function associated with a rigid sphere 

 moving away from the wall at unit velocity. The functions 

 4> , <|) -, will be constructed by means of the method of imagest 



2. Some theorems on im^iges * 



The theorems that follow supplement well known theorems 

 on images. See Milne -Thomson, Theoretical Hydrodynamics, 

 Chapter 15o 



