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Starting from the hypothesis that the carcinogenic effects of prenatal X-rays are a direct consequence of the mutagenic actions of ionizing radiations, and that the malignant diseases so caused (radiogenic cancers) have a different age distribution from other juvenile malignancies (idiopathic cancers), methods are developed for solving the problems posed by the limitations of the available data (Oxford Survey of Childhood Cancers). Explicit formulae for estimating the number of radiogenic cases in a simple situation are derived. The problems of truncated contingency tables and a simple approximate method for them are discussed. It is shown that in the situation described minimum modified chi-square estimation can be more effective for multinomials than maximum likelihood. It is demonstrated that the possibility of a memory effect in the records of X-raying can be discounted, and that radiogenic cases are genuine. It is shown that each diagnostic group has a characteristic radiogenic age distribution which differs from the corresponding idiopathic age distribution in being more sharply peaked and having a slightly older mean age. Small changes in the age distributions due to improved methods of diagnosis, etc., are demonstrated. Irregular changes in the cohort trend of the radiogenic hazard due to alterations in X-raying practice, etc., are shown.