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Summary Suppose that a series of street‐lighting posts has to be kept supplied with lamps. Is it cheaper to replace the lamps individually as they fail, or to replace all the lamps at regular intervals, replacing only those that fail between these intervals, and thereby to save labour but to sacrifice some lamps that have burnt only a short time ? That question, and some variants of it, is discussed in Part I. Precisely similar problems obviously arise when other systems are substituted for street‐lighting and other perishable members for lamps. In Part II computations are performed for one particular example. In Part III a subsidiary problem is discussed which is of no great practical importance, but leads to some remarkable algebraic identities.