But 



I 



P r 



dT = 



PQ 



•PQ 



V-v 

 ^PQJ 



dT 





dS 



(55) 



Substituting (55) and (54) now yields (53) for P in E. 



Next consider the case that P is in D, and let -¥■_ denote the volume 

 and S^ the surface of a small sphere of radius r^ about P. Then we have 



V„ X 



JVxv „ r Vxv „ r Vxv 

 — dT = V X dT + V X ^ 



dT 



(56) 



where D' = D - V^p^. The last integral in (6) is proportional to the 

 velocity induced by the vorticity within ¥■ , according to (52) and hence 

 must vanish as the radius of the sphere approaches zero. Also, by 

 Equations (54) and (55) , we have 



I 



V X I Z^ dT = V X I B^ dS + V X I I^ dS 

 ' '.. ^PQ ' i ^PQ ' i ^PQ 



+ V. 



v-v -^ dT 

 J . ^PO 



(57) 



22 



