Transparent Inactive Crystal Elates. 175 



1 /9l\ , • , 



or tg s sin r = — ( — I +cos r sin \j/ t 



q l \dxj 1 



(IT \ 

 j x J=a xl x\ + a li y\ + a l ,z\ 



which from (27) 



= — ( — a u cosr 1 sin </>,— a ls cos </*, +« ia sin /•, sin \f/ y \ 



Accordingly 1 



_ (g\ — « n )cos r, sin ¥>, — «„ cos ^) + « 13 sin r, sin y/, ( a ) (32) 

 ^ " 1_ ^ 2 j sin r, 



On eliminating ^ s from (29a) by means of (32) we find 

 sin (rj + rjfsin \j/ 1 sin y/ a cos (r 1 —r 1 ) + cos ^ cos f 2 ] + 



sin r, sin i/r r . . . , . , , 

 H L a u — ?"i) cosr i sm "Aj — « ]3 sin ^ sin i// 1 + a ]2 cos t/rj + 



sin r, sin ii, . ., , • , , , -, . 

 \ («»- <?,) cos r 2 sm ^i-«h Sln r 2 sm & + a„ cos yvj = 



which on rearrangement becomes 



■This expression was first derived by P. Kaemmerer (Neues Jahrb., Beil. 

 Bd. xx, 206, 1905), though by a method different from the above. 



2 From the equations (30) and (31) the following relations can also be 

 derived: 



—cos 1^1= — ( — ) = — ( — ai2 cos ri sin fi—a^ cos i/»i+ai 3 sin r x sin ipi ) 

 qi \dy) > q 2 i V / 



or tg 



an equation identical with (17). 



or tg^ = q '~ a22 (32a) 



a-n cos ri— a 23 sm ri 



Also coss! = - 



^I'\ Vl + tg* s, 



© 



; £g Si cosri + sin ri sin ^i 



—a, 3 cos ?'i sin Vi — a 23 cos ii>i + (a 33 — a 2 i) sin ?*! sin i/>, /o<y. \ 



or <CfSi = o (^ D i 



y g 2 ! cos » - i 



an expression for tg s, which is apparently novel. On equating (32) and 

 (326) we find 



t . _ an cos ri— ot 23 sin r t 



2 2 i— «n cos 2 rx— a 33 sin 2 ri +2ai 3 sinn cos n * c ) 



From (32a) and (32c), we have 



(g 2 i— a 22 )(g 2 1 — an cos 2 ri — a 33 sin 2 r, +2 ai 3 sin r 1 cos ri) 



=(a 12 cos r-j— a 23 sin r^) 2 



an expression free from V and identical with (16). 



