1884.] homogeneous polarized light seen in a uniaxal crystal. 83 



or sin r 2 cos (2rdr +...) + cos r 2 sin (...) 



k _ 2kcos20d0 

 ~ ± sin 20 + sin 2 20 ' 

 expanding this side by simple differentiation. 



IT 



Now since r 2 = 2n — 1 ^ , 



2 



A; 

 cos r 2 = and sin r 2 = + - — — n ; sin 20 = k 



~ sm 20 



_ =±i; ' 



~V 2 /sin20 -sin26> + k 2 



Whence cancelling terms we have as the relation between 

 dr and d0 



4*r 2 dr\ 

 dr 2 



'" rdd 



But if in figure PM be perpendicular to OQ 



PM = rd9, 



MQ = dr, 



dv 

 and p the radius of curvature = = — ^ . 

 r 2rd0 



Thus 2p = ^^?i. 



Therefore the curvature varies directly as the cube of the 

 distance r of the points of contact from the origin. 



Also we see if k = 0, p = oo , this gives us the flattened parts of 

 curves for k = 0. and serves to verify the formula. 



If k — 1, p = gives the point circles to which the ovals di- 

 minish when k = l. 



Proposition XVII. If tangents be drawn from the origin to all 

 ovals of intensity k the centres of circles of curvature at points 

 of contact lie on the unicursal quartic 



y x= ~2k-- 



6—2 



