If, for example, _Z_ is equal to unity divided by _t_ less one, the coefficient of the _x_th power of _t_ in the product of _V_ by _Z_ will be the coefficient of the _x_ + 1 power of _t_ in _V_ less the coefficient of the _x_th power. It will be then the finite difference of the primitive function of the index _x_. Then the character _Δ_ indicates a finite difference of the primitive function in the case where the index varies by unity; and the _n_th power of this character placed before the primitive function will indicate the finite _n_th difference of this function. If we suppose that _T_ be unity divided by _t_, we shall have _T_ equal to the binomial _Z_ + 1. The product of _V_ by the _n_th power of _T_ will then be equal to the product of _V_ by the _n_th power of the binomial _Z_ + 1. Developing this power in the ratio of the powers of _Z_, the product of _V_ by the various terms of this development will be the discriminant functions of these same terms in which we substitute in place of the powers of _Z_ the corresponding finite differences of the primitive function of the index. Now the product of _V_ by the _n_th power of _T_ is the primitive function in which the index _x_ is augmented by _n_ units; repassing from the discriminant functions to their coefficients, we shall have this primitive function thus augmented equal to the development of the _n_th power of the binomial _Z_ + 1, provided that in this development we substitute in place of the powers of _Z_ the corresponding differences of the primitive function and that we multiply the independent term of these powers by the primitive function. We shall thus obtain the primitive function whose index is augmented by any number _n_ by means of its differences. Supposing that _T_ and _Z_ always have the preceding values, we shall have _Z_ equal to the binomial _T_ - 1; the product of _V_ by the _n_th power of _Z_ will then be equal to the product of _V_ by the development of the _n_th power of the binomial _T_ - 1. Repassing from the discriminant functions to their coefficients as has just been done, we shall have the _n_th difference of the primitive function expressed by the development of the _n_th power of the binomial _T_ - 1, in which we substitute for the powers of _T_ this same function whose index is augmented by the exponent of the power, and for the independent term of _t_, which is unity, the primitive function, which gives this difference by means of the consecutive terms of this function.