## §22.8(iii) Special Relations Between Arguments

In the following equations the common modulus is again suppressed.

A geometric interpretation of (22.8.20) analogous to that of (23.10.5) is given in Whittaker and Watson (1927, p. 530).

Next, let

Then

For these and related identities see Copson (1935, pp. 415–416).

If sums/differences of the ’s are rational multiples of , then further relations follow. For instance, if

then

is independent of , , . Similarly, if

then

Greenhill (1959, pp. 121–130) reviews these results in terms of the geometric poristic polygon constructions of Poncelet. Generalizations are given in §22.9.