I. It is shown that a general argument can be given for omitting the diverging parts of any quantized field interacting with a particle. The field equations thus freed from all singularities still contain the reaction of the field on the particle to the same extent as the reaction can be derived classically from the conservation of energy. The new field equations can be solved generally in terms of a simultaneous set of integral equations and do not give rise to any fundamental difficulties whatsoever.

II. The new field equations are applied to the multiple processes of the meson theory which occur, for instance, when a meson collides with a nuclear particle, the meson splitting up into a number n of secondary mesons. The cross-sections γn are worked out for n = 1, 2, 3, 4. γn has a maximum at some energy εn which increases with n. For ε ≫ εn, γn decreases like ε−2n−4 except for n = 1, when γ1 ∞ ε−2. γ1 (ordinary scattering) is larger than all the other γ's at all energies, but the probability of high multiplicities is comparable with that for low multiplicities or even greater in some energy regions.