290 _44. 



Proof * If P is any point, the Stokes stream 

 function "^ (see Milne-Thomson, Chapter 15, esp. pp. 420, 

 421) is 



T — /x-i-^ (r-r,) ^ ^^ (r^-rg) ^ /^(rg-r,) , 



If P is on the sphere, we have r-, = — ^ r, , 



OQ, ^ 



' a / 



Tn = Tr, ( a property of inverse points, due to the 



similarity of triangles OPQ, , OQ,P), and consequently 



■yj/" = - u — -^-^ = const. Therefore -^^ = on the sphereo 

 Combining this result with the known image of a point 

 source (Milne-Thomson, pp. 420,421), we obtain the following: 



Theorem 2 . The image of a point source of strength m 

 situated, at the point Qo, together with a uniform line sink 

 of strength jx per unit length extending from Q. to Q^,, 

 where m = MQiQo (so that the total strength is zero), is 



the following: a point source of strength m at the 



^2 oQt 

 point Qo and a uniform line sink of strength M "~a '^^'^ 



unit length extending between Q,, Qg* 

 5» The construction of <|>g^ « 



The function ^ is required to satisfy the boundary 



conditions (3). If the plane were not present, ^ would be 



a2 2 



5— , which represents a source of strength A placed at 0. 



The presence of the plane, however, causes the boundary condi- 

 tion (3) on the plane to be violated. To satisfy it, introduce 

 the image of the source relative to the plane, which is a 

 source of equal strength at the reflected point 0-. But the 

 boundary condition on the sphere is now violated, and to 



