sentences:  The change in the particle positions r"k" for an infinitesimal change in the "D" generalized coordinates is
  
  Taking two derivatives with respect to time yields an equivalent equation for the accelerations
  
  The work done by an infinitesimal change "dqr" in the generalized coordinates is
  
  where Newton's second law for the "k"th particle
  
  has been used.Substituting the formula for "d"r"k" and swapping the order of the two summations yields the formulae
  
  Therefore, the generalized forces are
  
  This equals the derivative of "S" with respect to the generalized accelerations
  
  yielding Appell's equation of motion
  
  Euler's equations provide an excellent illustration of Appell's formulation.Consider a rigid body of "N" particles joined by rigid rods.