Finally, when P is a "smooth" point of S, i.e., a point at which the 

 tangent to S is continuous, we introduce a small hemisphere about P, of 

 radius r , and apply the Poincare transformation to the so-diminished 

 volume D' , and to the bounding surface, consisting of the hemispherical 

 surface S_ and the remainder S' of S. The proof is similar to that for 



Figure 8 



P in D. In Equations (57) and (58), we need only to replace S by S', and 

 in Equation (59) Attv becomes 27Tv , or, indeed, c 

 of S of solid angle a. Instead of (61) , we need 



in Equation (59) Attv becomes 27Tv , or, indeed, av if P is a corner point 





lim V„ X I ±L^ dS = V„ X I S^ dS 



and 



i, PQ Jr, PQ 



lim V dS = V ~ dS 



rg-O J , ^PQ { ^PQ 



which are verified by observing that the velocity fields of the surface 

 distributions of vorticity, n x v, and of sources, v • n, are continuous. 



Equivalent Irrotational Flow From Poincare Transformation 



Let us apply the Poincare transformation (53) to a case in which the 

 flow is entirely due to the vorticity in a domain D, bounded by a closed 

 surface S. We shall seek to express the velocity at a point P of E as the 

 gradient of a velocity potential. 



24 



