of the exponents or indices x and x´ of these powers; this function I shall call the primitive function of which V is the discriminant function.
Let us multiply V by a function T of the two variables t and t´ developed like V in ratio of the powers and the products of these variables; the product will be the discriminant function of a derivative of the primitive function; if T, for example, is equal to the
variable t plus the variable t´ minus two, this derivative will be the primitive function of which we diminish by unity the index x plus this same primitive function of which we diminish by unity the index x´ less two times the primitive function. Designating whatever T may be by the character δ placed before the primitive function, this derivative, the product of V by the nth power of T, will be the discriminant function of the derivative of the primitive function before which one places the nth power of the character δ. Hence result the theorems analogous to those which are relative to functions of a single variable.
Suppose the function indicated by the character δ be zero; one will have an equation of partial differences. If, for example, we make as before T equal to the variable t plus the variable t´ - 2, we have zero equal to the primitive function of which we diminish by unity the index x plus the same function of which we diminish by unity the index x´ minus two times the primitive function. The discriminant function V of the primitive function or of the integral of this equation ought then to be such that its product by T does not include at all the products of t by t´; but V may include separately the powers of t and those of t´, that is to say, an arbitrary function of t and an arbitrary function of t´; V is then a fraction whose numerator is the sum of these two arbitrary functions and whose denominator is T. The coefficient of the product of the xth power of t by the x´ power of t´ in the development of this fraction will then be the integral of the preceding equation of partial differences. This method of integrating this kind of equations seems to me the simplest and the easiest by
the employment of the various analytical processes for the development of rational fractions.
More ample details in this matter would be scarcely understood without the aid of calculus.
Considering equations of infinitely small partial differences as equations of finite partial differences in which nothing is neglected, we are able to throw light upon the obscure points of their calculus, which have been the subject of great discussions among geometricians. It is thus that I have demonstrated the possibility of introducing discontinued functions in their integrals, provided that the discontinuity takes place only for the differentials of the order of these equations or of a superior order. The transcendent results of calculus are, like all the abstractions of the understanding, general signs whose true meaning may be ascertained only by repassing by metaphysical analysis to the elementary ideas which have led to them; this often presents great difficulties, for the human mind tries still less to transport itself into the future than to retire within itself. The comparison of infinitely small differences with finite differences is able similarly to shed great light upon the metaphysics of infinitesimal calculus.
It is easily proven that the finite nth difference of a function in which the increase of the variable is E being divided by the nth power of E, the quotient reduced in series by ratio to the powers of the increase E is formed by a first term independent of E. In the measure that E diminishes, the series approaches more and more this first term from which it can differ only by quantities less than any assignable magnitude.
This term is then the limit of the series and expresses in differential calculus the infinitely small nth difference of the function divided by the nth power of the infinitely small increase.