of the Rhomhohedral System. 177 



measured angles of a scalenohedron, it contains a smaller error 

 than any other equation. 



The figure represents the stereo- 

 graphic projection of some of the 

 principal poles and planes of a 

 rhomhohedral crystal, together with 

 the poles P of a form {hkl} to be 

 determined. The poles r are {10 0}, 

 o (1 1 1) ; therefore the poles b and a 

 are {2 1 1} and {011} respectively. 

 Let P be (hk I), P y , P /y the corre- 

 sponding faces repeated over b 

 and b ir Then P, is (h I k), and 

 P yy (khl). Let Q, P, it be the intersections of the pairs of 

 zones [PP y ] [Ob'], [PP yy ] [06 /y ], [OP] [&&„] respectively. 

 Then the indices of Q are (2h, k + l, k + l), of R (h + k, h + k, 

 2Z),andof7r(2A-&-Z, -h + 2k-l, -h-k + 2l.) 



The anharmonic ratio of the poles a, 6 y/ , 7r, b gives 



sin&7r sin 66^' r&7rl rbb,7\ k—l 



2h-k-l 



sm 7ra 



sin b. 





(Miller's i Treatise on Crystallography/ p. 14). Hence 



tan 6tt= tan XOP= ffi l )^ t 



Hi — k — l 



(i) 



The anharmonic ratio of the poles 0, r } Q, b gives, in a similar 



manner, 



tanOQ _ rOQn fOrl _ 2h-k-l 

 tan Or - UqJ : Vbr\~ 2{h + k + iy 



and writing D for the element Or, we have 



2h-k-l 



tan OQ= - 



2(h + k + l) 



tan D. 



(2) 



Similarly from the poles 0, R, b in andz (2 2 1) the dirhombo- 

 hedral face of r {l we obtain 



tan OR _ rRO "I r^Cn _ h + k-2l 

 tan D ~ |_R6 yy J : UU "" 2(A + ife + ' '" " ( ^ 

 From the right-angled triangle POQ we have 



tan 0P= tan OQ sec&7r; .... (A) 

 .-. from (1) and (2), 



__ (k-iy + (i-hy + (h-ky 



tan ur- 2(h + k + l) 2 tan ^ * W 



