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The matrix theory of electric machines is based on the assumption of magnetically linear iron. This paper removes the linearity assumption at the start, while permitting its reintroduction at a later stage as a computational simplification. A definition is given for inductance in the nonlinear case that leads to single-valued continuous inductances, with reciprocal mutual inductances. The resulting differential equations are solved to give a method for determining the inductances associated with any machine. The torque of a slip-ring machine is then expressed in terms of its inductance matrix. The expression is shown to reduce to the usual result in the special case of a linear machine, and to predict, with a high degree of accuracy, the torque of a small heavily saturated experimental machine.