Refraction  of  Light  by  intensely  Opaque  Matter.         327 
to  introduce  two  auxiliary  variables,  defined  by  the  equation 
C€> 
cos 
whence 
*<=\A-^ 
c2cos  2u=l 
c2sin2w= 
cos  2a  sin2  0   ~\ 
R2 
sin  2a.  sin2  0 
Thus 
ai  _  sin  0    fMceiu  _  Rce''(M+a> 
a       cos  0    sin  0  cos  0 
?  ~cos0  +  Rcel>+a)' 
If  this  quantity  be  called  P,  and  that  obtained  from  it  by  chan- 
ging the  sign  i,  Q, 
Hi  
where 
tanc?: 
P-Q 
The  intensity  of  the  reflected  light  is  therefore 
PQ  _  CQs2  e  —  2Rc  cos  °  CQS  (u  +  a)  +  R2c2 
^  ~  cos2  0  +  2Rc  cos  0  cos  (u  +  a)  +  R2c2 
=  1 
Re  cos  (u  -f  a) 
cos  0 
RV2 
-1  + 
Re  cos  (w  +  a) 
cos  0 
1  + 
cos2<9 
1+ 
_RV 
cos20 
■=*•('-& 
if 
cot/=  cos  (w  4-  a) 
Re 
cos  0 
1+ 
R2c2 
cos2<9 
=  cos  (u  +  a)  sin  2  tan-1  (  -n — )• 
:  change  of  phase;  its  tangent  is 
/cos  0  \ 
tand=  sin  (u  +  u)  tan  2  tan-1  (  -rr — ). 
c?  represents  the  change  of  phase;  its  tangent  is  given  by 
'cos# 
