1. End-plate potential (e.p.p.s) were recorded with a surface electrode from frog neuromuscular junctions blocked with high Mg and low Ca to study post-tetanic potentiation (potentiation). 2. The magnitude of potentiation was not directly related to the number of conditioning impulses, but was a function of the frequency and duration of the conditioning stimulation. 3. Potentiation was always greater following an equal number of impulses delivered at a higher frequency of stimulation. 4. Plots of the magnitude of potentiation against the number of conditioning impulses would sometimes show an upward inflexion depending on the parameters of stimulation. 5. These experimental observations were described by a model based on the assumption (1) that potentiation is linearly related to a residual substance, R(t), which accumulates in the nerve terminal during repetitive stimulation, and (2) that each nerve impulse adds an identical increment, r, of this residual substance. The data were not described by assuming a 4th power relationship between potentiation and R(t). 6. The upward inflexion in potentiation (see paragraph 4) is described by the model as resulting from an increase in the time constant for the decay of potentiation as the magnitude of potentiation increases. 7. The increment of residual substance r added by each impulse was independent of the amount of transmitter released during the conditioning train. This increment typically increased transmitter release by amount 1% of the control level in the absence of potentiation. 8. Suggestions are given to explain why potentiation of transmitter release, which is thought to arise from an accumulation of Ca-2+ in the nerve terminal, can be described assuming a linear relationship between potentiation and R(t), the proposed substance responsible for potentiation, under experimental conditions in which a 3rd to 4th power relationship would be expected to exist between external Ca concentration and evoked transmitter release.