380 A. C. LANE — THE ANGLES OF CRYSTALS. 



an equation of condition, which may be much simplified if <i..,_ -^- a^, in 

 which case we may drop the subscripts and write o.-, for then A — - 0, B = 

 — (sin (fj cot a (?i + m), C = 2 shi <p cot o., D ^= — cos <p^ cot a 

 {n, — m), E ^^ m — n, and F =^ — sin (f^ cos (p^ (n + m). Therefore — 



B^ -f- D^ = co(' a (ii^ -\- in^ — 2 cos 2 cTj m ») ; 

 CB = — 2 sin^ (f^ coi- o. (n + m) ; and 

 EBD = sin (f^ cos cTj co(" a (n + m) (n — m)'. 



If we substitute these values in (2), and cancel and reduce and replace 

 (1 — cos 2 (f^) by sin^ cr^, we shall have this formula, which connects m, n, <f^ 

 and a — 



+ m '^ 



2 ..2.,. ^.'.-^C^^f (3) 



This may also be derived directly from spherical triangles. If o^ and a, 

 which depend on the terminal faces, are known, then m and n can vary only 

 within certain limits. The point whose coordinates are (m~^ ir^) will in 

 fact be limited to an ellipse (figure 1) whose intercepts will be (dz sin 2 

 (fjcot a, 0) and (0, ± sin 2 <pJcot a). The extremities of its axes will be at 



the points \ , f (fjcot a, (fjcot ay A slight modification gives a simi- 



lar ellipse connecting the apparent angles of termination of microliths, when 

 it is their edges that we see instead of traces of their faces. 



The use of this theory may be shown by the application of it to another 

 example ; and with this we close. A characteristic accessory of the slates 

 and kindred rocks of northern Michigan is tourmaline. . It occurs in small 

 distinctly hemimorphic prisms. One end is inclined to be more bluish, and 

 is terminated with the basal plane or else very bluntly. The other end is 

 browner, and has a sharper termination. The most important, and therefore 

 most likely to be observed, terminal faces are rhombohedra. The funda- 

 mental rhombohedron of Rosenbusch (i?) has a basal Qdigo. angle of 133° 10' 

 and is Dana's — ^. It is represented by P^P.^P.^ of figure 3. The funda- 

 mental rhombohedron (B) of Dana is O^O.fi.^ For the rhombohedron q R 

 equation (3) will become (since cos 2 (p^ will then be cos 120° = — h, and 

 sin 2 (f^ will equal j/f , while q c will, according to Dana's Textbook (page 

 72), be equal to -|/f tan «) 



n~'^ + m~'^ -}- m~^ n~^ = (q c)l (4) 



Every value of q for different rhombohedra will give a different ellipse 

 upon which the point (m~\ 7i ') should lie. Now, we remember that ni"^ is 

 the tangent of the angle from the trace of the basis to the trace of P.^, or. 



