Refraction  of  Light  by  intensely  Opaque  Matter.  325 
in  the  displacement  or  strain,  are 
dt_aj, 
&"~*"     dx~  dx' 
when  x  =  0.     The  system  of  waves  is  given  by 
y  _V  J(a,x+by  +  ct) 
where 
2tt        /,      ,      2tt  .'  2ttV      2tt 
#=  — cost/,     o=  — sina,     c=  — —  = 
A,  A,  A,  T 
The  coefficients  b,  c  are  necessarily  the  same  all  through ;  the 
multipliers  f,  £",  (J  are  complex.      The  boundary  conditions 
being  the  same  as  for  transparent  media,  we  have  (Phil.  Mag. 
t"      a  — a 
August  1871)  ~f  = ',  only  that  now  al  is  not  real.     In  fact 
b        a~\-  a^ 
7i  n 
if  =pr=y,  p-  =7y,  we  obtain  from  the  differential  equations 
c2  =  72(a2+62),  1 
whence 
^«*rW+«'l, 
"We  see  that  at  is  determined  as  a  function  of  a  and  Z>  by  an 
equation  of  precisely  the  same  form  as  for  transparent  media, 
the  only  difference  being  that  (a2  is  now  no  longer  real.  If  we 
suppose  Ql  to  be  defined  by  the  equation 
sin  0.  =  —  sin  6. 
we  may  use  the  forms  previously  investigated. 
From  the  physical  interpretation  of  ft2,  we  see  that  its  real 
part    is  positive,    and  imaginary  part   negative.      If  we    write 
IT 
/x2  =  R2e2l'a,  2a  must  lie  between  0  and  —  ■=.     This  remark  will 
be  found  to  be  of  great  importance.  For  instance,  the  assump- 
tion by  Cauchy  and  others  of  a  real  negative  value  of  y?  in  their 
treatment  of  the  so-called  longitudinal  waves  produced  when 
light  vibrating  in  the  plane  of  incidence  is  reflected  from  the 
surface  of  transparent  matter,  really  corresponds  to  an  unstable 
medium  in  which  the  forces  resulting  from  a  displacement  tend 
still  further  to  increase  it. 
