It was shewn, that, if the ultimate sums of the digits of the terms of any arithmetic series, whose difference is prime to the local value of the notation employed, minus 1, be taken, such sums will range, without any recurrence, through all the digits of the notation, and then recur in the same order as before. It was then shewn, that in any integer series formed upon a given law, the sums of the digits of the terms will have a fixed period of recurrence. This was proved in polygonal and figurate numbers, in the series of squares, cubes, &c., in the successive powers of a given root; in the series whose general term is m.m+1…m + r−1; and in that whose general term is xm + a xm−1.…+l.