An attempt has been made to clarify the cone formation mechanism of liquid-metal field-emission electron sources. The relation between the apex radius and the external field is calculated with the equations of balance between the electric field and surface tension stresses, field emission current density, and space charge limited current with small charge approximation. As a result, two solutions for the apex radius were found for a given external voltage for lithium. For gallium, also, two solutions were obtained but these solutions appeared at a limited range of the external voltage. The presence of two radii can be interpreted in that the balance between the electric field and the surface tension stresses is achieved with and without the presence of large space charge. The process of initiation and termination of cone formation is explained by the threshold voltage at a given apex radius and by the solutions of the above equation at the threshold voltage. The present calculation showed that surface tension stress easily balances with electric field stress for metals with a lower work function such as lithium. (C) 1999 Elsevier Science B.V. All rights reserved.