The statistical theory of nuclear fission was originally developed to deal with binary fission. An attempt is made to extend the theory to the case of ternary fission. The probability of a fission mode, given that the nucleus divides into three fragments, is first calculated. Attention is thereafter focused only on the probability distribution of fission modes representing the various possible allocations to the fragment translational motion and internal excitation of the energy available for both translation and excitation. This leads to ternary-fission distributions of the initial kinetic energy (and to complementary distributions of the initial excitation energy) of the fissioning nucleus. Our results show that the ternary-fission distributions of the initial kinetic energy are wider with peaks at higher values of kinetic energy than the corresponding binary-fission distributions obtained previously. The (spontaneous-) ternary-fission distributions are peaked at energies of the order of 1 MeV and have full-widths-at-half-maximum (FWHM) in the neighborhood of 3 MeV. These figures are of experimental interest, since they may be used to predict the energy spectra of the fission fragments in ternary fission.