It follows from symmetry that the two solutions indicated by the alternative signs in (6.12) lead to the same method. Thus there exists a unique two-stage implicit Runge-Kutta method of order four, defined by

(6.13)

a method originally proposed by Hammer and Hollingsworth.

Note: The method we have derived above extends to general R. For any , there exists an R-stage implicit Runge-Kutta method of order .