Comment on “A Laplace Transform Proof of the Theorem of Moments for the Instantaneous Unit Hydrograph” by M. H. Diskin

respectively in the express:•on [Kendall and Stuart, 1958]. In the case of the expansion of the Fourier and Laplace transforms, U/ must be equal to the coefficient of (ip)Vr! and (p)Vr!, respectively. Therefore, definition of the transform of Q or R by a polynomial in p of order n is equivalent to the definition of Q or R by the first n moments. A slightly different way of looking at the relationship is followed by the author, and also by March and Eagleson [1965], in which it is shown that the first n derivatives of the transform at p = 0 are the first n moments about the origin. This implies that the definition of the IUH by its first n moments is equivalent to describing the • = O(p) relationship by the first n derivatives at p = 0. Obviously, provided that n is large enough or that the form of • = O(p) is sufficiently restricted, this definition is adequate. If n is large, Taylor's expansion could be used to find explicit values of O(p) from •(0) and the derivatives of • at p = 0. Similarly, if the form of U = U(t), and hence that of • = O(p), is restricted and contains a limited number of parameters, specification of a limited number of derivatives at p = 0 is sufficient to define U over all p's. Consider as an example. the two-parameter gamma distribution, equation 3a [Nash, 195,7] approximating the IUH