The determination of the probable effect, upon the increase of population, of the extinction of small pox, is a problem which has been discussed at some length by three of the great continental mathematicians of the last century, namely, D. Bernouilli, D'Alembert, and Laplace. I propose, first, to solve the problem by what may be termed the theory of the composition of decremental forces, and then to reproduce (by way of comparison) from Mr. Todhunter's History of the Theory of Probabilities, the solutions given by the three eminent mathematicians in question.