The mathematical theory, of the magnetomotive force developed by armature windings is developed and presented in a new manner, and in such a form that it is at once directly applicable not only to, the more, common cases but also to abnormal cases, such as unbalanced windings, and to pole-changing windings, etc.An attempt has been made to present the subject matter in simple mathematical language and to give the various series concerned in as simple a form as possible. In general, the conclusions arrived at iiu the various sections of the paper have been succinctly stated.The method of treatment adopted is to start from the Fourier series representing the M.M.F. distribution due to a single coil, and to build up from this series the corresponding series for any complete winding. The effect of the value of the coil-span is discussed, it being shown that, in general, a single coil gives rise to an M.M.F. distribution corresponding to the joint effect of all numbers of poles from 2, 4, 6, etc., upwards. The effect of the spread of the coil-side is briefly discussed, and data relating to the various winding factors concerned are given in a simple graphical form applicable to harmonics of all orders, whether odd or even.The resulting M.M.F. due to a number of coils carrying the same current is then deduced from the effect of a single coil, heteropolar arrangements also being discussed.In the sections dealing with alternating-current windings, a similar method is employed. It is shown that all types of polyphase windings fall simply into one category, the treatment being perfectly general and directly applicable to all types of windings, whether normal polyphase, full-pitch or chorded, or a winding intended for pole-changing. The effects of higher harmonics in the current wave are discussed, and data relating to the harmonics present in the M.M.F. developed by polyphase windings are given.