A practical means for estimating the average performance of an electron microscope is proposed. Some first-order theory of the operation of an electron-microscope objective is given to demonstrate the extreme sensitivity of the instrument to the adjustment of the illuminating system. It is shown that by considering the illumination as being produced by a two-lens system it is possible to explain qualitatively all the effects observed in practice in connection with the illumination of the specimen. Practical information regarding the exact adjustment of the illuminating system is also given. The cause and elimination of multiple images is discussed. Changes in the design of the electron source and the use of an interchangeable aperture in the condenser lens are shown to improve the average performance of the instrument considerably.