The use of the mathematical models focuses on data from 33 States and 6 years in the period 1950-65, which was marked by relatively large changes in execution rates. For each of the States and four different pairs of years, the study estimates the growth or contraction factor in the State's homicide level between the first and second years studied. This factor can be treated as a product of (1) changes in the State's pattern of punishment over the period; (2) demographic changes associated with total State population, ethnic composition of the population, and degree of urbanization of the population; and (3) a regional time-trend factor that reflects age distribution of the population, general economic conditions, quality of emergency medical care, violence in the media, and availability of guns. In applying the models to the data, the analysis leans slightly toward some deterrent effect from capital punishment. The data are too limited, however, to make such a conclusion decisive. The limited number of executions that have occurred over time in the United States make it difficult if not impossible to affirm that any model decisively explains how capital punishment influences homicide rates. Five references and the mathematical models are provided.