subroutine potent(x,y,th,curv,vf,n,dcut)
*     *******************************************************
*     obtain the normal velocity of the fluid at the interface
*     (x(i),y(i),i=0,...,n) : the collection of equal-distanced
*      points;
*     (th(i),curv(i),i=0,...,n): the tangent angles, and
*      curvatures of these points.
*     (vf(i),i=0,...,n): the inward normal velocity of the fluid.
*
*     alternative points are used to discretize the integral and
*     the singular limit of the integrand is not needed.
*     *******************************************************
      integer n,i,j
      double precision x(0:8192),y(0:8192),th(0:8192),
     +                 curv(0:8192),vf(0:8192)
      double precision st,ct,rsq,dlg,sa,coef1,dcut,sum
      common/darcyc/coef1
      common/salpha/sa
*     for odd numbered points:
      do 10 i=1,n-1,2
         vf(i) = 0.D0
         st = dsin(th(i))
         ct = dcos(th(i))
         do 15 j=2,n,2
            rsq = (x(i)-x(j))**2 + (y(i)-y(j))**2
            dlg = (x(i)-x(j))*st - (y(i)-y(j))*ct
            vf(i) = vf(i) + dlg*dcos(th(j))/rsq
 15      continue
         vf(i) = coef1*(2.D0*sa*vf(i)/dfloat(n) + 0.5D0*ct)
 10   continue
*     for even numbered points:
      do 20 i=2,n,2
         vf(i) = 0.D0
         st = dsin(th(i))
         ct = dcos(th(i))
         do 25 j=1,n-1,2
            rsq = (x(i)-x(j))**2 + (y(i)-y(j))**2
            dlg = (x(i)-x(j))*st - (y(i)-y(j))*ct
            vf(i) = vf(i) + dlg*dcos(th(j))/rsq
 25      continue
         vf(i) = coef1*(2.D0*sa*vf(i)/dfloat(n) + 0.5D0*ct)
 20   continue
      vf(0) = vf(n)

*     check the sum of vf:

      sum = 0.D0
      do 40 j=1,n
         sum = sum + vf(j)
 40   continue

      do 50 j=0,n
         vf(j) = vf(j) - sum/dfloat(n)
 50   continue

*     *************************************

      call kfilter(n,dcut,vf)

      end