Leur Application Aux Probabilités Et Spécialement a La Recherche Du Milieu Qu’il Faut Choisir Entre Les Résultats Des Observations

I have given, it has been around thirty years ago, in the Mémoires de l’Académie des Sciences1, the theory of generating functions and that of the approximation of formulas which are functions of large numbers. The first has for object the relations of the coefficients of the powers of an undetermined variable in the development of a function of this variable to the function itself. From the simple consideration of these relations there results, with an extreme facility, the interpolation of the series, their transformation, integration of equations in the ordinary and partial differences, the analogy of the powers and of the differences, and generally the transport of the exponents of the powers to the characteristics which express the manner of being of the variables. The theory of the approximations by formulas functions of very great numbers is based on the expression of the variables given by some equations in the differences, by means of definite integrals which we integrate by some very convergent approximations; and there is this of the quite remarkable, namely that the quantity under the integral sign is the generating function of the variable expressed by the definite integral, so that the theories of the generating functions and of the approximations by formulas functions of very great numbers can be considered as the two branches of the same calculus, which I designate by the name of calculus of the generating functions. This which Arbogast2 has named Method of separation of the scale of operations is contained in the first part of the calculus of generating functions, which gives simultaneously the demon-