602 
ME. E. T. GLAZEBEOOK ON THE EEEEAOTION OE PLANE 
perpendicular on the tangent plane at P, then O Y is, we know, the extraordinary ray. 
Let a and 6 he the principal refractive indices in direction 0 A and 0 B, and let 
OP=r, OY =y>. Since OP and OY are respectively the directions of the wave normal 
and the ray, the angle POY=q, and p=r cos q ; let the angle POA=i//. 
Then 
Therefore 
Also 
whence 
P 
sec* 
1 _..9.f cosS ^ , !sill2 Y 1 
p 2 [a 4 6 4 J 
i7 _ r 4.J c o s2 t ■ sin 2 ^ ] 
1 1 a 4 ~ r Z> 4 J 
1 cos 2 Y sin 2 yfr 
r 3 n 3 Ir 
tan q—r 3 sin ip cos r ft 
a 2 — Jr 
(ii) 
But since r is a radius vector of the surface of wave slowness, r— sin <£/ sin (j)", and 
we have 
^ ^ 
sin 2 <p" tan q= - sin 2 </> sin xfj cos \p 
Again X Y (fig. 1) =90°—i//, and from the triangle X B Y 
Also 
Thus 
sin XY=sin BX sin XBY, 
cos BX=cos XY cos BY, 
sin 3 f'tan q= 
or cos if ;=sin j8 sin (X -j- <j>") 
or cos /3= sin xp cos 6" 
sin 2/3 sin (A+ (p' f ) sin 2 </> 
cos Q n 
and equation (10) becomes 
tan #=tan /3 cos cos (<j>—<f>") 
ft 2 —& 3 sin 2/3 sin (X -f cp") sin 2 cp 
^ 2ft 2 & 2 sin (c p -f <p") cos 2 $" 
but 6" can be found in terms of e/>, (f>" and constants, from the formula 
tan 0 "—tan /3 cos (X+<£"), 
and 0 is thus expressed in terms of </>, cp" and constants. 
The value of a has been found already; it is the ordinary refractive index of the 
spar for the light used, and has been shown to be 1*662. 
b is the extraordinary index, and may be taken with sufficient accuracy for the 
purpose from Mascart’s or Budberg’s determinations. Either of these give 6=1*488 
as the value of the extraordinary index, corresponding to the value 1*662 for the 
ordinary. 
