56-57] SOURCES AND SINKS. 65 



Finally, we may imagine simple or double sources, instead of 

 existing at isolated points, to be distributed continuously over 

 lines, surfaces, or volumes. 



57. We can now prove that any continuous acyclic irro- 

 tational motion of a liquid mass may be regarded as due 

 to a certain distribution of simple and double sources over 

 the boundary. 



This depends on the theorem, proved in Art. 44, that if </>, <jf/ be 

 any two functions which satisfy V 2 </> = 0, V'ty' = 0, and are finite, 

 continuous, and single- valued throughout any region, then 



where the integration extends over the whole boundary. In the 

 present application, we take </> to be the velocity-potential of the 

 motion in question, and put </>' = 1/r, the reciprocal of the distance 

 of any point of the fluid from a fixed point P. 



We will first suppose that P is in the space occupied by the 

 fluid. Since <j/ then becomes infinite at P, it is necessary to ex- 

 clude this point from the region to which the formula (5) applies ; 

 this may be done by describing a small spherical surface about P 

 as centre. If we now suppose S% to refer to this surface, and BS 

 to the original boundary, the formula gives 



At the surface 2 we have d/dn (l/r) = 1/r 2 ; hence if we put 

 S2 = r 2 cfar, and finally make r = 0, the first integral on the left- 

 hand becomes = 4-7r(/> p , where </> p denotes the value of <j> at P, 

 whilst the first integral on the right vanishes. Hence 



d l rt\ 



(7). 



This gives the value of < at any point P of the fluid in terms 

 of the values of </> and d(f>/dn at the boundary. Comparing with 

 the formulae (1) and (2) we see that the first term is the velocity- 



L. 



