( 370 ) 



tg hZ = tg V cos {x — y). 



Now 



\ — tq aZ tq hZ 1 — t<i a Ui v cos (x -\- y) cos (x — y ) 



cot2y = cot{aZ-]-bZ)^ 4 ^ = ^^W-^^ ^W V • (1) 



^ ' ^ tg aZ + tg hZ tg n cm {x -^y) + tg v cos (x—y) 



As is indicated in the figure, here the pa rticuhir case is considered 

 that the zone-axis lies in the phine forming right angles with the 

 acute bisectrix, so that n -\- v z= jr. If we let the zone-axis succes- 

 sive! j make different angles a with OP, ^'arying from to :r, and 

 if, at the same time, we let this plane perform a revolution about 

 this axis, then JW passes through the whole surface of the globe and 

 conseciuentlv the extinction with regard to OZ can be calculated as 

 a function of a and x for each arbitrary section through the crystal. 



As {i -\- V =: Jt, the formula (1) can be simplified as follows : 



1 + <i/' f* <^os {x 4- y) cos {x — y) 



cot Zy rz: 



tg (I [cos {x -\- y) — cos (x — y)] 



from which we derive: 



— {cos^ (I + sin^ (I cos^ y) -\- sin^ n sm' x 

 cot 2y 



sm X 

 Now in A ZOA cos ft := sm OA cos a = sin V cos a 



tqPZ 



and in A ZFA cos ^ AZP= ^^- 



tgAZ 



or 



/rr \ tg a 



cosl ^ — r \=:smY.= —-. 



\2 J tg n 



If in (2) we substitute the values in a and V for n and y, 

 we get: 



1 — sin'^ {I sin^ y 1 *i/i"'' fi 



cot 2g z= ^ ; . -; \- — ; — . sin x = 



sin 2fx sin y sin x sin 2|n sin y 



1 — Ig'^ a co.s'^ fi 1 1 — cos'^ ft 



2 cos^ ft tg a sin x 2 cos^ [itg a 



1 — sin" V sin'^ (t 1 1 — si?i^ V cos^ a 



sin 2« . sin'^ V sin x sin 2a sm* V 



From tiiis form we can deduce what follows. For .r = 0, // 



