Placing _δ_ before the primitive function expressing the derivative of this function, which multiplies the _x_ power of _t_ in the product of _V_ by _T_, and _Δ_ expressing the same derivative in the product of _V_ by _Z_, we are led by that which precedes to this general result: whatever may be the function of the variable _t_ represented by _T_ and _Z_, we may, in the development of all the identical equations susceptible of being formed among these functions, substitute the characters _δ_ and _Δ_ in place of _T_ and _Z_, provided that we write the primitive function of the index in series with the powers and with the products of the powers of the characters, and that we multiply by this function the independent terms of these characters.
We are able by means of this general result to transform any certain power of a difference of the primitive function of the index _x_, in which _x_ varies by unity, into a series of differences of the same function in which _x_ varies by a certain number of units and reciprocally. Let us suppose that _T_ be the _i_ power of unity divided by _t_ - 1, and that _Z_ be always unity divided by _t_ - 1; then the coefficient of the _x_ power of _t_ in the product of _V_ by _T_ will be the coefficient of the _x_ + _i_ power of _t_ in _V_ less the coefficient of the _x_ power of _t_; it will then be the finite difference of the primitive function of the index _x_ in which we vary this index by the number _i_. It is easy to see that _T_ is equal to the difference between the _i_ power of the binomial _Z_ + 1 and unity. The _n_th power of _T_ is equal to the _n_th power of this difference. If in this equality we substitute in place of _T_ and _Z_ the characters _δ_ and _Δ_, and after the development we place at the end of each term the primitive function of the index _x_, we shall have the _n_th difference of this function in which _x_ varies by _i_ units expressed by a series of differences of the same function in which _x_ varies by unity. This series is only a transformation of the difference which it expresses and which is identical with it; but it is in similar transformations that the power of analysis resides.
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.