The linear quadratic equation for fractionated radiotherapy has already been adapted to include a time factor for tumour repopulation: loge cell kill (E) is given as a function of dose per fraction (d), number of fractions (n), overall treatment time (T) and the clonogen doubling time (Tp). By incorporating a normal tissue isoeffect and replacing the relationship between T and n by a function f, the equation for E can be rewritten as a more complex function of d. In this form, E and d are continuous variables so that the dose per fraction (d') required to produce maximum values of E for isoeffective late normal tissue effects can be found by differential calculus. The derived equation takes the form (beta kTp-alpha Tp)d2 + 1.386fd + 0.693fk = 0 and when solved for d provides a direct estimation of the optimum dose per fraction. Where normal tissue sparing is possible and the tumour dose z is related to the normal tissue dose d, the optimum dose per fraction z' can be found by solving the equation (beta kTp-alpha gTp)z2 + 1.386fgz + 0.693fk = 0 The results show that a critical minimum dose per fraction is required to counteract rapid tumour clonogen repopulation in both conventional and accelerated radiotherapy. The calculus method is reasonably accurate for larger fraction numbers, when clonogen doubling times are 3.5 days or longer and for conventional radiotherapy given 5 days per week. The model is even more accurate for accelerated hyperfractionated radiotherapy providing that there is complete repair between successive fractions. Where greater normal tissue sparing is possible, as with focal teletherapy methods and brachytherapy, higher tumour doses per fraction can be used to increase further the tumour cell kill without exceeding normal tissue tolerance. These predicted doses per fraction are consistent with clinical experience when the given constraints in terms of frequency of treatment are considered. The model described can be used for tumours in which repopulation occurs at a constant rate throughout treatment. For tumours in which accelerated repopulation occurs, the optimum dose per fraction can be separately calculated for the initial phase of slow repopulation (for which very small doses per fraction are optimal) and also for the second phase of rapid repopulation (for which either accelerated hyperfractionated treatments or hypofractionated focal methods of treatment would be appropriate). The limitations of the model are fully discussed including the need for accurate radiobiological predictive assays. In the future such assays of pre-treatment doubling times and tumour cell radiosensitivities could be used to determine reasonable ranges for the optimum dose per fraction in experimental tumours and subsequently in clinical trails. This approach could produce major improvements in the therapeutic potential of radiotherapy.