326  The  Hon.  J.  W.  Strutt  on  the  Reflection  and 
The  value  of  fx,  is  Refa,  and  at  perpendicular  incidence  (&  =  0), 
a 
±fi,  say  +fi, 
as  the  sign  of  fju  is  in  our  power.  Now,  since  the  refracted  wave 
is  ^lrleiia'x+b^+ct\  wherein  x  is  negative,  we  see  that  the  real 
part  of  at  must  be  positive,  while  the  imaginary  part  is  negative. 
The  same  is  true  of  Re'a  ;  or,  as  R  is  taken  positive,  a  lies 
between  0  and  —  — .     Since,  as  we  have  seen,  2a  is  situated  in 
IT 
the  same  quadrant,  a  must  lie  between  0  and  —  — .      The  value 
of  a  is  determined  by  tan2a=  —  ^— .  It  vanishes  with  h, 
and,  on  the  other  hand,  when  h  :  6Dt  is  very  great,  approximates 
to  —  —.     In  this  extreme  case  the  real  and  imaginary  parts  of 
fju  are  numerically  equal;  the  imaginary  part  is  never  the  greater*. 
I  have  been  thus  particular  to  examine  the  limits  between 
which  a  may  lie,  because  it  appears  to  me  that  there  is  on  this 
point  a  serious  omission,  not  to  say  error,  in  Eisenlohr's  paper. 
In  that  investigation  the  necessity  of  a  limitation  on  the  magni- 
tudes of  the  real  and  imaginary  parts  of  //,  does  not  appear, 
mainly  because  the  author  has  assumed  at  starting  expressions 
for  the  incident,  reflected,  and  refracted  waves  without  reference 
to  the  differential  equations  tacitly  implied.  To  suppose,  as  he 
does  for  silver,  that  a  =  83°,  and  therefore  2a  =166°,  is  tanta- 
mount to  the  assumption  of  a  medium  essentially  unstable f. 
We  may  now  proceed  to  transform  the  expression  for  the  re- 
flected wave 
\[ ~~ a  +  a, 
In  terms  of  0,,  — = yr,  where  sin  0.=  -  sin  6.    To  simplify 
'  a       tan  0/  p 
the  expressions,  it  is  convenient,  following  Cauchy  and  Eisenlohr, 
*  I  apprehend  that  this  conclusion  is  not  limited  to  the  particular  form 
of  the  differential  equation  which  has  been  assumed.  Whatever  that  may 
be,  fi2  will  always  consist  of  a  real  and  an  imaginary  part,  of  which  the 
former  cannot  be  supposed  negative  without  compromising  the  stability  of 
the  medium. 
f  In  Eisenlohr's  paper  the  incident  wave  travels  in  the  direction  of  the 
positive  x,  while  I  here  suppose  the  opposite.  The  change  amounts  to  a 
reversal  of  the  sign  of  c ;  and  thus,  on  Eisenlohr's  supposition,  the  real 
part  of  /x3  ought  to  be  positive  and  the  imaginary  part  also  positive ;  his 
result  requires  that  the  real  part  should  be  negative. 
