Polarization in Biaxial Crystals. 367 



There are black dots given by 



1^ = 0/2 -f ft7T, 



, D l + k\ 0f 

 tan — = tan zy 



= jr- tan 2i/r. 



Hence as the analyser is turned round, a black dot makes 

 its appearance at the axis when = D/2, and then divides 

 into two which trace out the curve given by the second 

 equation. This curve is {vide infra) the Airy's spiral for the 

 case of biaxial crystals. 



If two plates of equal thickness but of opposite rotatory 

 powers are superposed between crossed nicols, the intensity 

 is proportional to 



. 2 D/l-#Yr 2k or • D ■ •-. D-l* 



m w\nrp) Li+i^ cos2 ^ sm T- sln2 ^ cos -2j ' 



and hence, in addition to the ordinary Gassini's ovals, we have 

 the Airy's spirals given by 



tan — = tan 2yfr 



Z Z A' 



= .— tan 2f ; 

 or, if we confine our attention to points near the axes, by 



— = Zy-f-7l7T. 



z 



To trace the spirals, we draw the Gnssini's ovals first (it is 

 convenient to arrange that the lemniscate be included in them) 

 and mark the hyperbola which would be the brushes if there 

 were no rotation. The spirals are to be drawn so that they 

 cross the brushes or ovals only at their intersection, and so 

 that the sense of the rotation is the same in the spirals about 

 either axis. Near to an axis the spiral resembles one of the 

 two perpendicular spirals seen in quartz ; further off the spirals 

 envelope both axes. 



7. We can modify the above work so that it applies to 

 bodies where the rotation is due to the structure of the mole- 

 cules or to their arrangement. For replace Yr cr in the first 

 equations by Vyo\ On proceeding to plane waves, y becomes 

 — w/>A, and 7 becomes a vector proportional to \. This makes 

 no other change necessary, and w r e need only replace 7 in the 



