A special problem involving the passage of a beam of gamma rays through air is examined. The effect of Compton scatterings on the sharpness of the gamma‐ray beam is discussed. Other mechanisms whereby the beam is softened are not included because the interest is only in the component of the beam which suffers little energy loss.The problem of interest is similar, in some respects, to multiple scattering of electrons with negligible energy loss. The mathematical expression for the collision process is different, however. A method used by Fermi in regard to multiple scattering of electrons is also used here. The Klein‐Nishina cross section for Compton scatterings is introduced into the present problem in order to obtain scattering probabilities.Some numerical results are obtained which can be applied to cobalt‐60 gamma rays. The results are presented in graphical form.