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PYRAMIDAL SYSTEM. 21 eight triangles by great circles through the poles of the forms 001 and no, the poles of Xhkl will be found in four alternate triangles, and the poles of the form Xkkl in the remaining four alternate triangles. 69. The form bounded by all the faces of the form hkl, in which the order of h, h is the same or different, according as h, k have the same or different signs, is said to be hemihedral with P-allcl faces, and will be denoted by the symbol nhkl, where , 18 the symbol of any one of its faces. The symbols of the faces of the form nhkl are contained in the first and third columns; those of nkhl in the second and fourth columns of the table in (05). If the surface of the sphere of projection be divided into eight lunes by zone-circles through the poles of the forms 100, 110, the poles of nhkl will be found in four alternate lunes ; and hose of 7rkhl in the remaining four alternate lunes. VO. The form bounded by all the faces of the form hhl, in which ie order of the indices h, h is the same or different according as an odd number of the indices are positive or negative, is said 0 be hemihedral with asymmetric faces, and will be denoted by ie symbol ah/cl, where hkl is the symbol of any one of its aces. The upper and lower halves of the table in (05), contain ^ l0 faces of nhkl, akhl respectively. I he poles of ah/cl are the eight alternate poles of the form : > and the poles of akhl are the remaining eight alternate poles of the form hkl. Vl. The poles of the form hkl are symmetrically arranged with respect to each of the five zone-circles drawn through the poles of every two of the forms 001, 100, 110. The poles of Kfikl are symmetrically arranged with rdspect to the two zone- circles drawn through the poles of the form 001, and those of the form 110. The poles of Xhkl are symmetrically arranged with respect to the two zone-circles through the poles of the form 001, and those of the form 100. The poles of -nhkl are symmetrically arranged with respect to the zone-cirele passing through the poles of the form 100. V2. The two hemihedral forms, either with inclined or with parallel faces, derived from the same holohedral form, are identical in all respects, position excepted; for by making the sphere of projection revolve through two right angles round two opposite poles of the form 100, the poles of xhkl and nhkl come into the places of those of «,7ikT and nkhl respectively; and by making it revolve through two right angles round two opposite po es of the form no, the poles of Xhkl come into the places of ose of Xkhl. The two hemihedral forms ahkl, akhl are essen tially different.