If the function indicated by _δ_ placed before the primitive function is zero, we shall have an equation of finite differences, and _V_ will be the discriminant function of its integral. In order to obtain this discriminant function we shall observe that in the product of _V_ by _T_ all the powers of _t_ ought to disappear except the powers inferior to the order of the equation of differences; _V_ is then equal to a fraction whose denominator is _T_ and whose numerator is a polynomial in which the highest power of _t_ is less by unity than the order of the equation of differences. The arbitrary coefficients of the various powers of _t_ in this polynomial, including the power zero, will be determined by as many values of the primitive function of the index when we make successively _x_ equal to zero, to one, to two, etc. When the equation of differences is given we determine _T_ by putting all its terms in the first member and zero in the second; by substituting in the first member unity in place of the function which has the largest index; the first power of _t_ in place of the primitive function in which this index is diminished by unity; the second power of _t_ for the primitive function where this index is diminished by two units, and so on. The coefficient of the _x_th power of _t_ in the development of the preceding expression of _V_ will be the primitive function of _x_ or the integral of the equation of finite differences. Analysis furnishes for this development various means, among which we may choose that one which is most suitable for the question proposed; this is an advantage of this method of integration.
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.