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An interesting extension of Waring’s famous problem is the following:— Can every sufficiently large n be expressed, as the sum of s almost equal k-th powers; or, more generally, can every sufficiently large n be expressed as the sum of s positive k-th powers almost proportional to s arbitrarily assigned positive numbers Xl5 x2, ... x, ? I have developed two methods to discuss this problem, one based on the Hardy-Littlewood method for the solution of WARING’s problem and the other on the new VINOGRADOFF method* for the solution of the same problem. In this paper I shall discuss the case k3 by the first of these methods. The case of five or more squares may be treated in the same way, and the results are similar; somewhat deeper and more troublesome analysis is required to deal with the case of four squares. The principle of the method employed here is that of weighting ” the various representations of n as the sum of s k-th powers in such a way as to make predominant the particular representation of which we are in search.