which occurs in all theories of numerical functions hitherto considered. The two most highly developed theories of this kind are those in which multiplication in the ring of all numerical functions is abstractly identical with C (Cauchy) or D (Dirichlet) multiplication of infinite series.t Lehmer's five postulates are sufficient for the development of a theory of inversion as exemplified in the cases of C, D multiplication, without requiring, as is the fact in those cases, that the function ?(x, y) (his {t(x, y)) of composition be a polynomial. For C multiplication, q (x, y) _x+y -1, instead of the usual x+y, as a change in notation justifies; for D multiplication, 0(x, y) _xy. But these are not the only ?(x, y) which give an arithmetical theory of composition (as in the papers cited); to mention only three further instances, there is von Sterneck's "L.C.M. calculus," quoted by Lehmer, in addition to the well known compositions ?(x, y) M(x, y), where M is either "max" or "min." It is of considerable interest then to see precisely what position is occupied in the general theory of composition developed in the paper (B) by the classical theories in which multiplication is abstractly identical with either C or D. We shall prove that if (p(x, y) is a polynomial in x, y, then, in order that the composition + (x, y) = n, where n is an arbitrary constant integer >0 and x, y are variable integers >0, shall lead to an arithmetical theory of composition, it is necessary and sufficient that + (x, y) be either x +y 1 or xy, namely, that multiplication in the ring of all numerical functions be either C or D.