1. In the gnomonic projection of the poles of a crystal on any plane the projection of the pole of (hkl) is the point (hkl), and the projection of the zone [HKL] is Hx + Ky + Lz = 0, a suitable triangle of reference and suitable triliteral coordinates being taken.

2. By means of this theorem simple proofs of well-known crystallographic theorems may be obtained.

3. Some relations between the number (u_r) of zones of a crystal containing r faces, and the number of faces (v_r) lying in r zones may be found; e.g., n(n−1)=∑2nr(r−1)ur, ∑2n{(r−1)vr−1−rur}=0 where ∑v_r = n .