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A new describing function called the Elliptic Describing Function (EDF) is defined which allows the waveshape of a limit cycle to be determined after the amplitude and frequency have been found. The application of the EDF is nonrigorous mathematically but is justified by examples. The technique is based on the fitting of the nonlinear characteristic-whatever its shape so long as it is single-valued, odd-symmetrical, and frequency-independent-by an odd cubic polynomial. Considerable evidence exists which justifies this seemingly drastic approximation. Three examples are given illustrating the errors involved. Comparisons are made with the ordinary DF and new RMS DF for two examples and with a total of six DF's for a third. The results are encouraging.