Elementary Geometry to Crystallography. 



851 



This equation remains true when K is no longer a point in the 

 axis OX, and is no longer the origin. 



OF OP_ sinKDOsinKRO 

 OD ,0R ~ sin KFO sin KPO " 



Let PQ, PRj PS denote the angles which the faces P, Q, R, S 

 make with the face P, supposed to be all measured in the same 

 direction. Then 



sinPQ sin (PR -PS) 



sin KDO sin KRO 



sin KFO sm KPO '"' siu(PR-PQ) sin PS ' 

 Hence 



sin(PR-PQ) sin PS = z sin (PR -PS) sinPQ. 



12. PS is given in terms of PQ,PRj and the indices of P,Q;R,S 

 by equations easily deducible from the above. These are 



i(cot PS - cot PR) = cot PQ- cot PR 

 and 



fsinPQtan^=sin(PR-PQ), tan(PS-iPR)=taniPRtan(^|-^j. 



V w 



The indices of S may be obtained by eliminating - and - 



between either of the last equations and m< + vi; + w?<; = 0, where 

 UV1V is the symbol of the edge formed by the intersection of any 

 two of the three faces P, Q, R. 



13. Sin(PR-PQ)sinP = fsin(PR-PS)sinPQ is readily 

 transformed into 



cos(2PR-PQ+RS) = (l-z)cos(PQ + RS) + zcos(PQ + RS), 



the upper or lower sign being taken according as PR is greater 

 or less than PS. This equation gives PR when the angle PQ 

 has been observed in one crystal, and the angle RS in another; 

 or when, as not uufrequently happens, the crystal consists of two 

 individuals in positions not accurately parallel, one of which has 

 the faces P, Q, and the other the faces R, S. Fig. 7- 



14. Let efg, pqr be the symbols of 

 any two edges OP, OR; KF, KR any 

 two edges in the face POR ; hkl, uvw 

 the symbols of any two faces contain- 

 ing the edges KF, KR respectively. 

 Then, since the edges efg, pqr meet 

 the face hkl in D, F, and the face uvw 

 in P, R, 



OF.OP = i.OD.OR. 



Draw DU parallel to KR, meeting OR in U, and FV parallel 



