Projection of the Sphere in Crystallography. 



41 



8. The expressions for u, v, w in terms of h, k, I, p, q, r, when 

 uvw is the symbol of a great circle passing through the points 

 hkl, pqr, are the same as the expressions for u, v, w in terms of 

 h, kj 1, p, q, r, when uvw is the symbol of the intersection of the 

 two great circles hkl^ pqr. These expressions are easily remem- 

 bered if we observe that, when written in a line, the last letter 

 in each of the six products is obtained by writing j9, q, r twice 

 in inverse order. A rule for forming them was also given in the 

 Philosophical Magazine for May 1857. 



9. Let efg be the symbol of the 

 great circle K P meeting B C, C A, 

 AB in D, E, F respectively ; pqr the 

 symbol of the great circle KR ; hkl, 

 uvw the symbols of any points Q, S. 

 Draw the great circle QS meeting 

 BC in N, and KP, KR in P, R. 

 Draw the great circles DQ, DS meet- 

 ing the great circle B E in T, V. 

 Draw the great circles KQ, KS ; and 

 suppose the great circles CT, CV 

 drawn. ^ ^ 



The symbols of A, B, C are 1 0, 1 0, 1 respectively ; 

 therefore (77) the symbols of BC, CA are 1 0, 1 0. There- 

 fore (t) the symbols of D, E are Og— f, gO — e. The symbol 

 of BE is eOg, the symbol of DQ is -(f/t + g/) ih gh, 

 and the symbol of DS is ~{{v + gw) iu gu. Hence (i), the 

 symbol of T is fg/i g(eA + fA; + g/) —eih, and the symbol of V is 

 igu g{eu + fv + gw) — efw. 



By (8), ir), 



sinPQ sinNS sin ET sin BV sin ECT sin BCV 

 sin NQ sin PS 

 By {e), 



sin BT sin EV ~ sin BCT sin EC V 



fsh 



fgM 



j^Qrp _ gjeh + fk + gl) 

 b 



sin BCT, 



.ECV=S(^-^+^+8:^)sinBCV. 

 a 



Hence 



sin PQ sin NS _ u eh + (k + gl 

 sin NQ sin PS ~heu-\- fv + gw' 

 In hke manner 



sin RQ sin N S _ m p/t -J- qA + r/ 

 sin NQ sin RS ~~ A pw + qv -f rw ' 

 Hence 



sin PQ sin RS _ di + {k + gl pw + qv + vw 



sin KQ sin PS ~ ])h + qk + vl cuThT-fgw' 



(«) 



