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



By writing ^ — cr^ for c we can refer c' to any initial line, and if we have 

 a similar eqnation derived from C, S, and P., we can eliminate r and have 

 an equation of the fourth degree to determine cr. This I have already ob- 

 tained/'^ If, in figure 3, (p is measured from O.^C, the bisectrix of 1\CI\; if 

 a^ = C'Pi, a, = CF,, <f, = I < P^CP,, H = cot < SO : SP,, and m = cot < 

 iS'C : /SPp then cp is determined by the following equations :' 



A cos -]- B ^ . E sin <p cos(p -\- F /-. 



tan <r = — Y^ 27-^ — .. , and sin z=z — . 1^/ f • — ; v^ ^ 



( cos -j- V A cos<p -{- C sm ip ' 



tan' {F- + D' - C) + ^a?r c (2 P'^ + Z>-' - 0" + P'^ - A' + ^'0 + 

 ^•^ _ ^4'^ _|- jp-^ _]_ 9 (/rr?i^^ c. 4- ^oy^ cr) (DB -}- EF - A C) = 0, 



where — 



^4 =:^ — COS cTj (cot (\ — cot <i..^ ; 

 B = — sm c-j (?i eo^ <i.^ -\- m. cot « J ; 

 C = 6-i>?. cTj (^cot (\ + <^o^ «.,) ; 

 D = — cos cr^ {)i cot «j — m cot a.^ ; 

 J^^ =^ m — n; and 

 7^' = — sin cTj eos cTj. 



Application to Augite, etc. 



§ 7. There is one most frequent case that we may consider in detail : It 

 arises when one of the three planes whose traces are supposed to be seen 

 bisects the angle between the other two. This occurs in hornblende, augite, 

 olivine, feldspar, titanite, pyrite and other minerals. It may be treated 

 by either construction ; and equation (5) of § 6 takes the much simpler 

 form — 



where- 



cos 2 cf = a -\- h ±. \/{a H- 6 + !)■' — 4 b, (1) 



a = I cot a and = I cot a ) . 



\n — m J \n — m J 



We have also — 



b {i — cos 2 ^0 = 1 + ^os 2 cr. (2) 



We derive (1) from (5) of § 6 as follows : cos a^ = 0, sin c^ = — l,cot <p^ = 

 cot «2 ; a-nd we may drop the subscripts and write cot « for both. Therefore, 



*Tschermak's Min. u. Pet. Mitth., October, 1887, p. 207. 



