1877.] seen through a Crystalline Plate. 395 



— = — = x%x-\-z%z= — bp. 

 x z b r 



Substituting in (6) and putting y* = 2j)p', we find for the curvature 



_ b(a 2 sin 2 + c 2 cos 2 + b 2 — a 2 - c 2 ) 

 9 b\a 2 sin 2 + c 2 cos 2 0) - aV ' 



or 



>~i _ g I (& 2 ~ « 2 ) cos 2 + (V - c 2 ) sin 2 6 1 m 

 9 c 2 (6 2 — « 2 )cos 2 + « 2 (6 2 -c 2 )sm^ ' ' ' ' ^ 



which gives the apparent index for light polarized in the principal plane 

 when a line in that plane is brought into focus. 

 Next let P lie in the elliptic section, then 



which reduces (6) to 



2{x 2 + z 2 - b 2 )(a*xtx + c 2 zlz) + {5 2 2 + z 2 - a 2 - c 2 ) + a*c 2 }y 2 = 0, . (8) 



and 



lx _ lz _ — p a 2 a?3# + c 2 2$z 



^~c^~ V(>V + cV) = aV+cV ' 



which, on putting p' for y 2 +2j?, reduces (8) to 



(a? + z 2 - 6 2 )(aV + cV)l - { 6 2 (^ 2 + z 2 - a 2 - c 2 ) + a 2 c 2 }p' = 0. . (9) 



We have also 



ci 2 x 



tan = — 



c 2 z 



which, combined with the equation to the ellipse, gives 



x 2 +z 2 



:a 4 x 2 -{-c i z 2 



sin 2 cos 2 a- 2 sin 2 6>+<T 2 cos 2 6> n a" 4 sin 2 + <r 4 cos 2 0' 

 On substituting in (9), and reducing, we find 



>-i _ {( b2 - O c°s 2 Q + (& 2 - c 2 ) sin 2 6 } (a 2 cos 2 + c 2 sin 2 0)* 

 9 a 2 (6 2 -a 2 )cos 2 + c 2 (6 2 -c 2 )sin 2 > • ( 10 ) 



which gives the apparent index for light polarized perpendicularly to the 

 principal plane, when a line in that plane is brought into focus. 



To sum up. For the pencil which is polarized in the principal plane 

 the apparent index for a line perpendicular to that plane is the real index 

 6 -1 or while for a line in the principal plane it is given by (7). For 

 the pencil which is polarized perpendicularly to the principal plane the 

 apparent index for a line perpendicular to that plane is given by (2), and 

 for a line in the plane by (10). 



On examining the expressions (7) and (10) for the radii of curvature 



