Observatio arithmetica

we exhibit, such that

is.
This series (g.) will itself also be periodic, just like the series (r.) of residues, and indeed

is its period. In which matter it is to be noted that the final terms of the period will always be gb-1 = d, gb = c, since, on account of rb = 0, it must necessarily be that rb-1 > rb < rb+1.
Therefore, having discarded the two final terms gb-1, gb, which are found to be the same in every period, we shall call the prior part

the principal part of the period P. Since, however, rμ+1 - rμ = rb-μ - rb-μ-1, it must be that gμ = gb-μ-1, i.e., g1 = gb-2, g2 = gb-3, and so on; consequently, the principal part of the period is always symmetric, whence the Π itself returns unchanged if the order of the letters is entirely reversed.

Example I. Let a = 4, b = 11; the series (r.) will be adorned with the signs c, d:

and hence the period

and its principal part

Example II. So that there may be an example of broader scope, let a = 7, b = 24; the period of the series (r.) will be

thence the period of the series (g.)

and its principal part