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PYEAHIDAL SYSTEM. 23 77. Lot n, k, l be tbe distances between any two poles of the form hkl, differing only in the signs of h, k, I respectively. Let r be the distance between any two poles having the indices h, h in different order, and the signs of the first, second, and third indices in one, the same as the signs of the first, second, and third indices in the other. Let o be the distance between any two poles having the indices h, k in different order, the signs of the first and second indices in one, different from the signs of the first and second indices in the other, the sign of the third index being the same in both ; and let m be the distance be tween any two poles differing only in the order of the indices K k, and in the sign of one of them. Then k I tan ip = _, tanlL = -cotEcosA, h h sin-JjK = cos \ l sin <f>, sin 11 = cos^lcos^, sin|o = cos | l sin (45°+ <f), sin ^E = cos cos (45°+ <f), cosm = (sin-^L) 8 . 78. If the distance between two poles of either of the forms «&o, hoi, hhl bo given, tlio distance between the two poles, or its supplement, will be one of the arcs f, k, l, whence, from the expressions in (74), (75), (76), the indices may be found. If the distance between any pole of the form hkl and each of two other poles of the same form bo given, the three poles not being in a great circle, the given distances, or their supple ments, will be two of the arcs ir, k, l, f, a, m, which being known, <p and l, and thence, from the expressions in (77), the indices may be found. 79. Let a, n, c bo the poles of loo, oio, ooi respectively; p tbe pole of hkl, Q that of pqr. Let q be in the zonc-circle pa. Then tan pa _pk _ pi tanQA hq hr Let (j be in the zone-circle pn. Then tnnrn _ ql _ qh tail q i! kr kp Let q be in the zone-circle rc. Then tan PC rh rk
