Let us conceive now that _V_ be a function of the two variables _t_ and _t´_ developed according to the powers and products of these variables; the coefficient of any product of the powers _x_ and _x´_ of _t_ and _t´_ will be a function 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 _n_th power of _T_, will be the discriminant function of the derivative of the primitive function before which one places the _n_th 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 _x_th 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.