The generality of analysis permits us to suppose in this expression that _n_ is negative. Then the negative powers of _δ_ and _Δ_ indicate the integrals. Indeed the _n_th difference of the primitive function having for a discriminant function the product of _V_ by the _n_th power of the binomial one divided by _t_ less unity, the primitive function which is the _n_th integral of this difference has for a discriminant function that of the same difference multiplied by the _n_th power taken less than the binomial one divided by _t_ minus one, a power to which the same power of the character _Δ_ corresponds; this power indicates then an integral of the same order, the index _x_ varying by unity; and the negative powers of _δ_ indicate equally the integrals _x_ varying by _i_ units. We see, thus, in the clearest and simplest manner the rationality of the analysis observed among the positive powers and differences, and among the negative powers and the integrals. 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.