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()a n (n)[y(n) +1]* [y(n) + kn -1J (3 ) = E k=-1k+2k+-k. . ,kZ,k$O n1 kni Thus the partition k = lki+2k2+ +Ikt of k into ki ones, k2 twos, and so on, is weighted in a certain way by the product shown, and various partition functions are obtained by choosing the function 'y(n) appropriately. For example, if 'y(n) =1 for all n, then ay(k) becomes the unrestricted partition function; if 'y(n) = n for all n, then a,,(k) becomes the plane partition of Wright [2];' if 'y(n) =1 for rth powers and 0 otherwise, then a.,(k) is the number of partitions of k into rth powers; and so on. The purpose of this paper is to establish the asymptotic formula

which is equivalent to g2'(u-o-a(2u)/a_4(U), can be used to prove various results involving partition functions. Dobbie [3] recently constructed simple direct proofs of (1) and (2) that require no knowledge of elliptic functions; incidentally (2) can be derived from (1) by dividing by x -y and then letting y-*x. The writer [2] showed that by means of (2) one can give a very concise proof of the familiar formula for the number of representations of an integer as a sum of eight squares or of eight odd squares. In the present note we obtain the formulas for four and six squares in a similar manner (see for example [6, p. 307]).