## Some Applications of the Gnomonic Projection to Crystallography

Harold Hilton

Fellow of Magdalen College, Oxford

Communicated by Professor Miers

University College, Bangor, North Wales

**Summary:** 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* .