The utility of differential forms for understanding the origin of the self-dual gauge-field equations is illustrated by deriving the systems of linear partial differential equations introduced by Belavin and Zakharov and used in a different form by Ueno and Nakamura. The integrability condition for these systems of equations is then used to show their relation to a generalized form of the Ernst equation.

*Physics Essays* **3**, 406-413 (1990)