Landwebev 



the velocity at P induced by vortex elements at points Q of S, 

 and, by (41) , the negative of the_ second term is the contribution 

 from the local vortex eleraent Yp. Here Tp- is the length of the 

 chord joining the points P and Q of S and Vp denotes the gradient 

 with respect to the coordinates of P. 



The integral in (44) is not suitable for numerical evaluation 

 in the given form because rpQ, which goes to zero as Q approaches 

 P, occurs in the denominator of the integrand. This singularity can 

 be eliminated, however, in the following manner. 



First take the cross-product of (44) by np to obtain 

 ir£[''<j''^p(i)] X"pdS„4(^pX7;^)Xn^=UTXnp. (45) 



Since, in the neighborhood of P, both y^ and Vp(l/rpJ = rpo/r^Q lie 

 very nearly in the tangent plane at Q, their cross-product is very 

 nearly parauLlel to nQ,_and hence the integrand of (45) is proportional 

 to the angle between n and n or rpo/R, where R is the radius 

 of curvature of the arc of S subtended by the chord PQ, Thus the 

 order of the singularity of the integrand of (45) has been reduced to 

 that of i/rpQ. 



In order to elinainate this singularity, consider the relation 



PQ 

 since -y ♦ np = 0, Also we may write 



(47) 



since £ ^q • Vq (1 Ap(^ dS^ is the flux through S due to a sink of 

 unit strength at P. Applying (46) and (47), and noting that 



we obtain from (45) , 



466 



