Let s, k and n be positive integers and define rs,k(n) to be the number of solutions of the diophantine equation
in positive integers xi. In 1922, using their circle method, Hardy and Littlewood [2] established the asymptotic formula
whenever s≥(k−2)2k−1 + 5. Here , the singular series, relates the local solubility of (1.1). For each k we define to be the smallest value of s0 such that for all s ≥ s0 we have (1.2), the asymptotic formula in Waring's problem. The main result of this memoir is the following theorem which improves upon bounds of previous authors when k≤9.