Equations are developed for the rate of penetration of a substance into (or egress from) the zone included between two coaxial cylinders. In the most general case the penetrating substance is assumed to engage in diffusion, a chemical reaction of the first order and a zero-order reaction (e.g. metabolism). The rate of elimination of nitrogen from the tissues into the blood capillaries is considered as an example of diffusion alone, without any accompanying chemical reactions. Application of the simplified equations to this case strongly supports the view that the rate of elimination of nitrogen from the body when breathing oxygen is not conditioned by diffusion factors, but depends only on the volume of the various systems of the body and the rate of blood flow through them. The rate of passage of carbon monoxide from the blood to the myoglobin of red mammalian muscles is next considered as an example of diffusion accompanied by a first-order chemical reaction velocity. Calculations for resting and active muscles indicate that in the latter case certainly, and in the former case probably, the rate of uptake of carbon monoxide by the red muscles should be fast enough to affect appreciably the measurement of blood volume by the carbon monoxide method. Millikan's experiments on the rates of change of oxymyoglobin concentration in the resting soleus muscle of the cat are used as an example of the most general case, in which diffusion and chemical reactions, both of first order and zero order, are all involved. The application of the equations of this paper to his data are shown to lead to reasonable conclusions as to the number of open blood capillaries in the resting soleus muscle. Similar equations, with examples, are also developed for the case of a cylinder suspended in an infinite medium and for a sphere suspended in an infinite medium.