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with the nontrivial restriction on to that b = (Ta 2to)/d shall be a nonnegative integer, Ta denoting the trace 2so + js1 of a. This transformation of the problem uses a recent theorem of Pall and Taussky (see [12]) on the number of representations of a binary quadratic form as a sum of four squares of linear forms. Relatively little is known of the theory of reduction of quadratic forms over Rd, as compared with Z. As an example, it seems to be true that d = 5 and 8 are the only positive ring discriminants for which the form Hd = o + + 42 is in a genus of one class; but such results may be hard to prove, by availablemethods. Hence our transformation of the problem is advantageous. The restriction on the values of to is no handicap in the study of qualitative questions, concerning the mere representability as a sum of four squares in Rd. Hence we are able to obtain quite complete results of a qualitatative kind in Part I (cf. Niven [7]). Certain conjectures made by Cohn on the basis of machine calculations (see [1]) are confirmed. We might have simplified some details of proof if we had restricted Rd to be maximal, as some writers do. Such restrictions are unnatural: thus it is as significant that a totally positive number so + sl(3)1/2 (with s1 necessarily even) is a sum of four squares of numbers xo + xl(3)1/2 as the similar fact for so + s1co, Co = J(1 + (3)1/2). Among the simplest consequences of Part I is that the only values d for which every a satisfying the obviously necessary conditions is a sum of four squares in Rd are d = 5, 8, and 12, and the negative d's not divisible by 16. At the referee's suggestion we examined the remaining d's as regards sums of five or more squares,

Sums of four squares in a quadratic ring