100 Prof. Miller on the Anharmonic Ratio of Radii 



crystal, having given its symbol, and the 

 symbols and positions of three other ^j 

 sides ; or, having given the positions of 

 four sides of a face of a crystal, and the 

 symbols of three of them, the symbol of 

 the fourth may be found. 



Let the sides qr, pq be respectively 

 parallel to the axes of the zones efg, pqr ; 

 rp parallel to the axis of the zone con- 

 taining the face hkl ; pi waH rn parallel to 

 that of the zone containing uviv, meeting 

 qr, qp in /, n; 



9P _ Q^ ^P1 5^ _ sinjorg; sin qpl sin PKQ sin SKR 

 qn qr ~ qr pq sin qpr sin qlp sin QKR sin PKS 



Hence 



qp _ql _eh + ik-\-Ql pw + qi? -{- rw 

 qn~ qr "" pA + qA: + r/ ew + iv + gw ' 



(9) 



When the right-hand side of the equation is negative, the 

 points I, r or p, n lie on opposite sides of q. 



That the axes of any three zones may be taken for crystallo- 

 graphic axes was proved by Kupffer (Poggendorff's Annalen, 

 1826, vol. viii. p. 61, and Handbuch der rechnenden Krystallo- 

 nomie, p. 497). The expressions for the indices of any face when 

 referred to new axes, which I gave in ' Crystallography/ art. 28, 

 may be easily deduced from (7). 



Let efg, hkl, pqr be the sym- 

 bols of the zone-circles EP, FD, 

 DE ; X, Y, Z their respective 

 poles; G the pole of the face 

 mno ; P the pole of the face 

 uvw. Let u'v'w' be the symbol 

 of P when referred to the axes 

 of the zone-circles EF, FD, DE 

 as crystallographic axes. Then 

 (7), 



sinGFE sinPFD_em + fn-f go hw + kv-l-liw 

 sin GFD sin PFE ~ hm ■\- k?i + \o eu-\-iv + gw' 



sin GEF sin PED _ em + fw + go ])u + qv + rw 

 sin GED sin PEF ~ pm + qn + ro eu + i'v + gw ' 



