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An approximate integral recursion formula is developed which gives a relation between the probability that a photon will be transmitted through a slab of finite thickness with exactly k+1 collisions and the probability that it will be transmitted with exactly k collisions, where the latter probability is known for suitable ranges of slab thickness, of incident energy, and of incident angle. A similar formula also exists for the expected energy transmitted. The two recursion formulas have been used to calculate the transmissions with one, two, and three scatterings for photons incident on slabs of lead and of iron, and from these calculated transmissions the ``build‐up'' factors have been estimated. The build‐up factor for normally incident photons of 1 to 20 mc2 are given for lead slabs and for iron slabs having a thickness of 0 to 20 mfp. A second method, which considers the transmission through a thick slab as a succession of transmissions through thin slabs, is used to check the estimates of the build‐up factor obtained by the first method. This second method provides additional information in the form of the distributions of the transmitted photons. Finally, some transmission and reflection values for thin slabs of the Compton scatterer are presented along with some results for air which are obtained by the second (thin‐slab) method.