sentences:  The Jacobian of this system is the bordered matrix
  At regular points, where the unmodified Jacobian is full rank, the tangent vector spans the null space of the top row of this new Jacobian.Appending the tangent vector as the last row can be seen as determining the coefficient of the null vector in the general solution of the Newton system (particular solution plus an arbitrary multiple of the null vector).This method is a variant of pseudo-arclength continuation.