Projection of the Sphere in Crtjstallography , 45 



may be denoted by the symbols of the faces and zones they 

 respectively represent. ^ 



18. Let the straight lines four of 

 which pass through the point K, in- 

 tersecting the fifth in the points P, Q, 

 R, S, represent faces of a ci*ystal, 

 being the fixed point. Let efg, pqr 

 be the symbols of the zones the axes 

 of which are represented by the points 

 P, R, hkl, uviv the symbols of the 

 faces represented by the lines KQ, 

 KS. 



By («), 



PQRS 

 RQPS 



sin POQ sinROS 

 sinROQsinPOS* 



But OP, OQ, OR, OS are four zone-axes in one plane. Con- 

 sequently the corresponding zone-circles pass through one point. 

 The sine of the angle between any two zone-axes is equal to the 

 sine of the angle between the two corresponding zone- circles, 

 efg, pqr are the symbols of the zones of which OP, OR are the 

 axes ; hkl, uvw are the symbols of faces in the zones of which 

 OQ, OS are the axes. 



Hence by (a), (X,), 



[y) 



sinPKQ sin RKS _ PQ RS _ eh-{-ik + ^l ^u-^c{v + vw 

 sinRKQ sinPKS ~RQ PS ~pA-|-q/t-|-r/ ew+lT+gw' 



19. Let the symbols of the 

 faces represented by the lines 



AB, AC, BC, DE be given, and 

 let uvw be the symbol of the 

 face represented by the line 

 MN. By equating the value of 

 AD . BM : BD . AM, as given by 

 construction, with its value as 

 given by (v) in terms of the indices of the points A, B, and 

 those of the lines DE, MN, an equation is obtained of the form 

 au-\-bv + civ=0, where a, b, c are integers. The comparison of 

 the value of AE . CN : CE . AN, as obtained by construction, with 

 its value as given by (v) in terms of the indices of A, C, DE, MN, 

 yields a similar equation. By elimination between these two 

 equations the ratios of u, v, w may be found. 



20. Let the symbols of the faces represented by the lines AB, 



AC, BC, DE be given, and let u, v, iv, the indices of a face to be 

 represented by a line MN, be given. The numerical value of 

 AD . BM : BD . AM can be found by (v) ; knowing this, and the 



