1. Def. The reciprocal (1/N) of any number (N) will be termed a Binal, Tertial, Quintal, ... Decimal, ... r-mal fraction, when expressed in the scale whose radix is r = 2, 3, 4, ... 10, ... r respectively. In this paper the developments will be given for the most part in a general form in the r-scale; the examples will be in the scales of r = 3 and 5. The notation is similar to that of ordinary decimals. Thus (see Tables at end):