Conditions for the coefficients of Runge-Kutta methods for systems of n-th order differential equations

To derive order conditions for Runge-Kutta methods of Nyström or Fehlberg type, applicable to arbitrary order differential equations, a theory similar to that about Runge-Kutta methods for first order systems, due to Butcher 1], is developed. By a new definition of elementary differentials, which is independent of the order of the given system, each condition to be satisfied by the coefficients of the method directly follows from the representation of the corresponding elementary differential.