A model for the field-emitting electron gun is proposed in this paper. The cathode is a sphere on cone and the anode and tip holder are represented by a potential on a larger sphere. The potential everywhere is shown to be expandable in terms of Legendre functions of noninteger order. Some properties of these functions are discussed in an Appendix. The model gives equipotentials in the region of the tip with realistic shape and can also explain the variations in the total emission current as a function of tip-anode distance in good agreement with experimental data. Other models have realistic equipotentials near the tip but cannot take tip holders into account and cannot explain the emission data.