Now if we multiply the series of the development of _V_ by a function of the same variable, such, for example, as unity plus two times this variable, the product will be a new discriminant function in which the coefficient of the power _x_ of the variable _t_ will be equal to the coefficient of the same power in _V_ plus twice the coefficient of the power less unity. Thus the function of the index _x_ in the product will be equal to the function of the index _x_ in _V_ plus twice the same function in which the index is diminished by unity. This function of the index _x_ is thus a derivative of the function of the same index in the development of _V_, a function which I shall call the _primitive function_ of the index. Let us designate the derivative function by the letter Alembert placed before the primitive function. The derivation indicated by this letter will depend upon the multiplier of _V_, which we will call _T_ and which we will suppose developed like _V_ by the ratio to the powers of the variable _t_. If we multiply anew by _T_ the product of _V_ by _T_, which is equivalent to multiplying _V_ by _T²_, we shall form a third discriminant function, in which the coefficient of the _x_th power of _t_ will be a derivative similar to the corresponding coefficient of the preceding product; it may be expressed by the same character _δ_ placed before the preceding derivative, and then this character will be written twice before the primitive function of _x_. But in place of writing it thus twice we give it 2 for an exponent. Continuing thus, we see generally that if we multiply _V_ by the _n_th power of _T_, we shall have the coefficient of the _x_th power of _t_ in the product of _V_ by the _n_th power of _T_ by placing before the primitive function the character _δ_ with _n_ for an exponent. Let us suppose, for example, that _T_ be unity divided 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 _x_th power of _t_ in the product of _V_ by _Z_; this coefficient in the product of _V_ by the _n_th power of _Z_ will be expressed by the character _Δ_ affected by the exponent _n_ and placed before the primitive function of _x_.