0) 



normal to Four Faces of a Crystal in one Zone. 97 



- cos PX = 4 cos PY = - cos PZ, 

 e / 9 



7-COS QX = 7" COS QY = y COS QZ, 



n k I 



- COS RX = -COS RY = - COS RZ, 

 p q r 



- COS SX = - COS SY = — COS SZ. 



U V ' w ^ 



If D, E, F be any three points taken in order in a great circle 

 of the sphere, it is easily proved that 



■ ^,^^ [cos DX COS EY- cos DY cos EXl 

 sin DE *■ -' 



= ^-TVP [cos DX cos FY- cos DY cos FX] 

 sm DF *- -• 



= ^^ [cos EX cos FY - cos E Y cos FXl . 

 sin EF "- -• 



But P, Q, 11, S are in a great circle of the sphere ; therefore, 

 if we substitute P, Q, R and P, S, R for D, E, F in the prece- 

 ding equations, in such order as the case may require, and then 

 divide the equation thus obtained between sin PS and sin SR, by 

 the corresponding equation between sin PQ and sin QR, and 

 substitute for the ratios of the cosines their values in terms of 

 the parameters and indices, as given by equations (1), we get 



sinPQsinSR ek—fli uq — vp 



(2) 



sin QR sin PS ~ hq — kp ev—fii' 

 where the lower sign is to be taken when one only of the points 

 Q, S lies between P and R. 



Let efg be the symbol of any zone-circle passing through P, 

 pqr that of any zone-circle passing through R. The zone-circle 

 QS contains the poles Q, S. Therefore {' Crystallography,' 

 art. 13) its indices are hv — Iv, lu—hw, hv — ku. It intersects 

 the zone-circles efg, pqr in P, R respectively. Therefore [' Cry- 

 stallography,' art. 15), 



e = {[hv — ku) — g{lu — hw), 



f=g[kw—lv)—e{hv —ku), 



y=m{lu — hw) — {{kw — Iv), 



p = i{[hv — ku) — r{lu—hw), 



q = v{kiu — Iv) — p(Au — ku), 



r^\i{lu—/iw) — q(kw — Iv), 



ek -fh = [hv - ku) [eh +ik + g/) , 



hq~kp = [ku — hv) (p/t ■\-(\k + xl), 



ev —fu = [hv — ku) (cm -\-h + gw), 



uq — vp =■ [ku — hv) (pw -f qw -f rw) . 



Phil. Mny. S. 4. Vol. 13. No. 84. Feb. 1857. H 



