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



To this we can apply fornuil?e 105 to 108 of Chauvenet's Trigouometry, 

 rediiciug it ultimately to the form — 



2 sill Sin sin 2 (Cm — a) 

 tan 2x, = cos 2 (Cm —«)(! + sin' Sm) + cos 2 F (siV Sm — 1}" ^^^ 



Similarly, for x.-, — 

 tan 2 X., = 



2 sin Sin sin 2 ( Cm + «) /j\ 



eos 2 (Owi + «) (1 + siV /Saw) + cos 2 F (siri^ Sm — 1) 



Again, eliminating SA, SB, SB^, SA^ from equations (4) and (2) we have — 



/ — (/ sill (Cm — a — F) sin (Cm — a -\~V) sin g sin h .^n 



/' — a' sin ( Cm -{- a -\-V) sin ( Cm + « — F) siii e sin f 



From formula 119 of Chauvenet's Trigonometry, sin esinf= - , , 'f f 

 so that, using equations (3) and (1), we have — 



. sin 2 x^ tan Cm — a-\-V tan Cm — a — F 



sin e sin f = —• — o — = — ; (9) 



J sm Sm i^,, Cm — a — V-t- tan Cm — a-\-V 



and .similarly — 



sin 2 X., tan Cm + « -f- F /a/i Cm -j- a — F 



tan Cm -f- a — V -\- tan Gin + a -j- F 



sin q sin h = -^ t' '* — (10) 



^ sin Sm . ^> . Tr , , ^. . . -r^ / 



Now, substituting (9) and (10) in (8) and reducing with the help of form- 

 ulae (105) to (108), as we did for (6), we finally get— 



. ' — a sin 2 .^2. sin 2 ( Cm — a) 

 7''^^7' "" s'mY^j sin 2 (Cm + «) ^ ^ ^ ^ 



Then (6), (7) and (11) may easily be transformed into the followiug equa- 

 tions : 



(-08^ Sm cos 2 V 



= — 2. "p^ sin 2(Cm — a)-\-(l-h siir Sm) cos 2 ( Cm — a) ■ (1 2) 

 tan Zi .i-| 



= — 2 p^L^ sin 2 (Cm + «) + yl + sin' Sm) cos 2 (Cm -f «) ; (13) 



sin 2 X, . / — a" 

 tan 2 Cin _ sin 2 x^ y — a' "^ _ . , . 



tan 2 a "" ^2j;.. / ZZJ"'' i 

 a 7 — ^^ 



sm 2 a^i 



