27 



I (R x«)(r. • q xn)<^a = -( (R x «) (r. • A x n)^ 

 = -i3itfx (r.xA'R)nc?CT 

 = ^« X F (r. 



A- R)c?T 



i ''V- 



= !»;[« x (r. X A)] [66] 



3 



Using Equation [33], the last term of the integral in Equation [61] becomes 

 I <I>[tt» X (r X n)] c?a= <I>[w x (r^. x n)] c?a = w x (r^. x $nc?a j 



since R x n = 0. Using Equation [37], we have 



4>7 



J .Dl 



[w x(r xn)]<^a = i5[ia X (r. X A)] [67] 



3 



Substituting these results in Equation [61], we have 



M3 (i)^0 [68] 



MOVING SINGULARITIES 



The cases which have been discussed so far are (1) discrete singularities which are 

 fixed with respect to the body, and (2) continuous distributions of singularities. While these 

 cases include the most important applications, flows exist which can be discussed in terms of 

 discrete singularities moving within the body. In the present section, the analysis will be ex- 

 tended to include this case. 



The control surface S enclosing the moving singularity is taken to be a sphere with 

 center fixed at t Xt j, the instantaneous position of the singularity at time t^. At the time 

 t^ + 8t, the singularity will have moved to ^ (t + 8 t) or referred to the center of the control 

 sphere R (SO- 



