by t ; then in the product of V by T the coefficient of the x th power of t will be the coefficient of the power greater by unity in V ; this coefficient in the product of V by the n th power of T will then be the primitive function in which x is augmented by n units.
Let us consider now a new function Z of t, developed like V and T according to the powers of t; let us designate by the character Δ placed before the primitive function the coefficient of the xth power of t in the product of V by Z; this coefficient in the product of V by the nth power of Z will be expressed by the character Δ affected by the exponent n and placed before the primitive function of x.
If, for example, Z is equal to unity divided by t less one, the coefficient of the xth 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 xth 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 nth power of this character placed before the primitive function will indicate the finite nth 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 nth power of T will then be equal to the product of V by the nth 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 nth 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 nth 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 nth power of Z will then be equal to the product of V by the development of the nth power of the binomial T - 1. Repassing from the discriminant functions to their coefficients as has just been done, we shall have the nth difference of the primitive function expressed by the development of the nth 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.
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.