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GEHT3EA.Ii GEOMETEXCAXi PBOPEETIES OE CEYSTALS. 7 18. It frequently happens that a number of faces of a crystal intersect each other, or would intersect if produced till they met, in parallel lines. Such an assemblage of faces is called a ‘ zone.’ The faces of a zone are all perpendicular to one plane, and their poles lie in a great circle, which will he called the ‘ zone-circle.’ A line parallel to the intersections of the faces of a zone will be called the ‘ axis’ of the zone. 19. Let hkl, pqr be the symbols of any two faces in a zone. Then, if u = kr — lq, t =. Ip — hr, w z= hq — kp, uvw will be the symbol of the zone containing the faces hkl, Pqr, or of the zone-circle through the poles of hkl, pqr. 20. A face may be common to two zones, or its pole may be the intersection of two zone-circles. , Let hkl, pqr be the symbols of two zones. Then, if u — kr —■ lq, v = lp — hr, w = hq — kp, uvw will bo the symbol of the face common to the zones hkl, pqr, 21. Let uvw be the symbol of a face in the zone uvw. Then, uu + vi> + w«> = o. Any positive or negative whole numbers, including zero, which, when substituted for u, v, w, satisfy the above equa tion, are the indices of a face in the zone uvw ; and any positive or negative whole numbers, including zero, which, when substituted for u, v, w, satisfy the same equation, are the indices of a zone containing the face uvw. 22. Let i>, q, e, s be four poles in fig. 7. one zone-circle, i>e being larger than i‘Q, and pq, i>e, ps measured in the same direction from p. Let their symbols be,— p efg, q hkl, e pqr, uvw. [ps] _ fw — gv [se] vr — icq Czsl _ ~ .7* [gh] kr — lq Then, (cot PS — COtPll) gu — ew ev —fu 'I 1 8 "s 1 uq — ip ’ gh — el ek -fh 1 ' Sk 1 i hq — kp j3j(cotPQ - ■ COtPE). -