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OEXEBAL GEOMETEICAL PEOFEETIES OF CEYSTAES. points thus determined, one being in each of the three lines sox', toy", zoz'. 10. The lines xox', tot', zoz' will be called ‘axes;’ the portions OAj, OB!, oCj ‘ parameters.’ The symbol 321 will bo used to denote the face parallel to a 3F s €i, 214 the face parallel to a 2 b_ 1 o 4 , and generally, hlcl will be used to denote the face parallel to a^c,, where OA* = iOAj, OB„ = lOBi, OC, = jOC,, I being any positive or negative whole numbers including zero, and oa a , ob*, oc, measured towards x, y, z, or towards x > x , z', according as h, Jc, I are positive or negative. The numbers 3, 2, 1, and 2, I, 4 in the two former examples, and the letters h, Jc, I in the latter, will be called ‘ indices.’ An index taken negatively will be distinguished by a negative sign placed over it. The indices h, Jc, I, which by taking different integral values determine the positions of the different faces of the same crystal, are seldom large. The highest index does not com monly exceed six. Parallel and opposite faces have the same indices with different signs. 12. The linear dimensions of the faces are not subject to any law; consequently, a face of a crystal may be represented by any piano parallel to it, on the same side ot the point o. 13. IV hen one of the indices of a face is fig. 2. zero, the point in which the face intersects the corresponding axis is indefinitely dis tant, or the face is parallel to that axis. When two of the indices are zero, the face is parallel to the two corresponding axes. . In the parallelopiped dfg (fig. 2) having its edges parallel to the axes ox, oy, oz, uef, lqk are the faces 100, 100; gef, ldii are the faces 010, 0I0, and deg, iifk, are the faces 001, 001. , 11. In ox, oy, oz (figs. 3, 4, 6, e,) take on, ok, ol, respect ively proportional to Toa 1; ioiq, ioc,; oil = on, ok = ok, oE = on. . ^* 10n ; in the prism qe (fig. 3) having klkE for its section, and its jixis parallel to ox, qki^ ekE are the faces old, 0Id, and QkE, eki, are the faces old, old. In the prism having eiiEh (fig. 4) for its section, and its axis parallel to oy, use, iitE, are the faces hoi, hoi, and nsf, Iitl aro the faces hoi, hoi.