324  The  Hon.  J.  W.  Strutt  on  the  Reflection  and 
This  is  for  one  surface.  If  the  plate  be  quite  transparent, 
the  reflection  may  be  nearly  the  double  of  the  above.  Now  for 
glass  at  an  angle  of  45°,  Magnus  found  no  smaller  value  of  I 
than  '084;  and  as  this  must  be  attributed  almost,  if  not  entirely, 
to  the  first  surface,  it  is  clear  that  something  not  taken  account 
of  in  FresnePs  theory  must  have  come  into  operation.  But  by 
far  the  most  remarkable  result  was  with  fluor-spar  for  the  reflect- 
ing, and  rock-salt  for  the  radiating  plate.  The  reflection  at  33° 
was  no  less  than  '23,  at  45°  -242,  and  at  62°  '335 ;  and  to  this 
the  second  surface  cannot  contribute  sensibly.  Unquestionably 
therefore  the  reflecting-power  of  fluor-spar  for  a  certain  kind 
of  dark  radiation  is  greatly  in  excess  of  what  can  be  accounted 
for  without  an  extreme  opacity — a  result  which  is  the  more  re- 
markable because  for  dark  radiation  in  general  fluor-spar  is  one 
of  the  most  transparent  things  known.  The  reflection  from  a 
plate  of  rock-salt  was  found  to  be  much  the  same  as  from  glass; 
but  here,  I  presume,  we  may  consider  both  surfaces  to  be  ope- 
rative, in  which  case  the  result  is  normal*.  It  is  curious  that 
opacity  first  diminishes  the  reflection  from  a  plate,  and  then 
when  extreme  increases  it  again,  and  that  without  limit. 
The  effect  of  opacity  is  represented  mathematically  by  the  in- 
troduction into  the  differential  equation  of  a  term  proportional 
to  the  velocity.  If  we  suppose  that  x  =  0  is  the  surface  of  sepa- 
ration, and  that  the  vibrations  are  parallel  to  z  and  perpendicular 
to  the  plane  of  incidence  xy,  the  differential  equation  in  the 
opaque  medium  is 
">  dfi  +A  dt  -reW  +dy*  )' 
where  h  is  &  positive  constant  depending  on  the  opacity,  and  D 
denotes  the  optical  density.  In  the  upper  medium  h  =  0,  and 
the  equation  is 
dt2  ~  1)  \dx2  +  dyV' 
Both  h  and  D{  are  subject  to  extensive  chromatic  variations;  and 
the  equations  are  therefore  not  to  be  regarded  as  general  equa- 
tions of  motion  applicable  to  all  possible  cases.  It  is  probable, 
however,  that  they  represent  with  sufficient,  if  not  absolute  ac- 
curacy the  laws  of  motion  for  a  system  of  plane  waves  of  given 
period,  provided  that  suitable  values  of  h  and  D;  are  assumed. 
The  boundary  conditions,  according  to  Green  and  Cauchy  (Phil. 
Mag.  August  1871),  expressing  that  there  is  no  discontinuity 
*  According  to  the  experiments  of  Masson  and  Jamin,  the  transmission 
of  a  perfectly  transparent  plate  is  always  about  92  per  cent.,  whether  the 
material  be  glass,  rock-salt,  or  alum.  This  is  in  agreement  with  the  cal- 
culation in  the  text,  as  about  8  per  cent,  would  be  reflected. 
