The first term of (53) is seen to give the velocity v due to vortic- 

 ity in the form of the vector potential (52), as 4itv . Since the assumed 

 flow is solenoidal, the third term of (53) is zero. We then have, from 

 (53), 



r B^ dS - V, r i^ dS (62) 



4Trv = V 



VP P , . . , ^ 



"^ ' ^PQ J^ ^PQ 



This expresses the velocity in terms of that induced by a source distribu- 

 tion of strength v /4tt and by a vortex sheet of strength n x v = v t, both 



on S. Here we have expressed the velocity vector v = nv + sv , where s 



^ n s 



is the unit vector in the direction of the projection of v on the tangent 

 plant at Q. Then, putting n x s = t, we obtain the form given above. 



In order to express the field of the vortex sheet as the gradient of 

 a potential, let us define a function <))-. , harmonic in D, which on S 

 satisfies 



n X V = n X V, v = V(|) (63) 



We have then 



__i = 



ds Is 



on S (64) 



and hence, by integration along s, the values of ())^ on S may be presumed to 

 be given in terms of the known values of v . Thus the boundary condition 

 (64) sets up a Dirichlet problem for determining ({) . 



Since V x v =0, application of the Poincare transformation to v^ , 

 taking (63) into account, yields 



C - - /• V, 'n 



y ^PQ ^ J ^PQ 



V X i^-^ dS = V^ -^^^ — dS (65) 



P ' "PQ ^ 1 "PQ 



25 



