The mathematical relations between basal energy metabolism, brain size, and life span in mammals have been investigated. The evolutionary level of brain development, or encephalization (c), is a function both of brain weight (E) and of body weight (P) according to (formula; see text) Brain weight was found to be a linear function of the product of encephalization and basal metabolic rate. The oxygen consumption of the brain (Mbrain) is proportional to both encephalization and body weight according to (formula; see text) The ratio of metabolic rate in the cerebral cortex to that in the brain as a whole depends solely upon the degree of encephalization and is independent of the size of the animal. The maximum potential life span of a mammal was found to be proportional to the product of its degree of encephalization and the reciprocal of its metabolic rate per unit weight. Life span may be regarded as the algebraic sum of two components: (1) a deduced somatic component (Lb) inversely related to the basal metabolic rate per unit weight, and (2) an encephalization component (Le) related directly to the evolutionary increase of relative brain size.