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Let $Z(t)$ be the classical Hardy function in the theory of the Riemannzeta-function. The main result in this paper is that if the Riemann hypothesisis true then for any positive integer $n$ there exists a $t_{n}>0$ such thatfor $t>t_{n}$ the function $Z^{(n+1)}(t)$ has exactly one zero betweenconsecutive zeros of $Z^{(n)}(t)$.

On the higher derivatives of Z(t) associated with the Riemann Zeta-Function