normal to Four Faces of a Crystal in one Zone. 101 



But 



sin GFE _ cos GX sin PFD _ cos FY 

 sin GFD ~ cos GY ' sin FFE ~ cos PX' 



sinGEF cosGX sin FED cosFZ 



Hence, if 



sinGED~cosGZ' sin FEF ~ cos FX' 

 e'cosGX i'cosGY c'cosGZ 



e»i + f« + go hm + kra + lo pm + qn + ro' 



-r cos r A = TT cos r 1 = -7- cos r Z, 

 a' o' c 



where 



u' = eu + iv + ^w,-^. 



i^ = \iu-\-\.v + \w, V (10) 



If we substitute for e, f, g ; h, k, 1 ; p^ q, r their values in 

 terms of the indices of the poles D, E, F, in which the zone- 

 circles EF, FJ), DE intersect, the resulting expressions for u',v', w' 

 resemble those obtained by Frankenheim (Crelle's Journal fur 

 Matheniatik, 1832, vol. viii. p. 181), but are not identical with 

 them ; for though the notation for faces is the same in both, the 

 axes are different. 



To find u, V, w in terms of u\ v\ ?<;'. 



Let efg, hkl, pqr be the symbols of the poles D, E, F, in which 

 the zone-circles efg, hkl, pqr mutually intersect. The symbols 

 of the faces 100, 010, 001, when referred to the new axes, become 

 ehp, f kq, glr respectively ; and the indices of the zones contain- 

 ing every two of them will be 



kr-lq, qg-rf, fl-gk; 



Ip-hr, re-pg, gh-el; 



hq — kp, pf— qe, ek — fh. 



But these are e, h, p ; f, k, q ; g, I, r respectively. Therefore 



u-=eu' -\- hv' +pu/, ^ 



v=fu' + kv' + qw', I (11) 



w =ffu' + Iv' + '■w'. J 



To find the symbol of the zone uvw, when referred to the axes 

 of the zones efg, hkl, pqr as crystallograpbic axes. 



Let efg, hkl, pqr be the symbols of the faces common to every 

 two of the zones, not contained in the zones efg, hkl, pqr re- 

 spectively ; u'v'w' the symbol of the zone uvw when referred to 

 the new axes. The zone uvw contains the faces Ovvv, wOu, viiO. 



