156  Mr.  A.  B.  Porter  on  the 
Now  the  ordinary  theory  of  the  transmission  grating  shows 
that,  in  the  case  nnder  consideration,  the  amplitude  of  the 
light  in  the  central  image  is 
a -to 
and  that  the  amplitude  in  the  rath  spectrum  is 
A    .    mira  /t, 
—  sin 7 .     (4) 
ra7r       a  +  b 
A  comparison  of  (2)  with  (3)  and  (4)  shows  that  the  first 
term  in  (2)  represents  the  amplitude  in  the  central  image, 
while  the  coefficient  of  the  cosine  in  each  succeeding  term 
represents  the  sum  of  the  amplitudes  in  the  two  spectra  of 
corresponding  order.  It  thus  appears  that  a  diffraction- 
grating  performs  a  double  process  of  harmonic  analysis.  In 
the  first  place  it  analyses  the  incident  radiation  according  to 
wave-length  in  (.he  well-known  manner,  distributing  the  colours 
in  order  in  each  spectrum  ;  in  the  second  place,  as  shown  in 
equations  (2),  (3),  and  (4),  it  analyses  the  distribution  of 
wave-amplitude  in  its  own  plane,  distributing  the  Fourier 
components  of  the  amplitude  curve  in  order  among  the 
successive  spectra . 
3.  We  may  also  look  at  the  matter  from  another  point  of 
view.  The  curve  drawn  in  fig.  1  not  only  represents  the 
distribution  of  wave-amplitude  in  the  plane  of  the  grating, 
but  is  also  the  curve  which  shows  the  distribution  of 
transparency  over  the  surface  of  the  grating  itself,  the  axis 
of  x  representing  zero  transparency,  i.  e.  complete  opacity, 
and  the  height  A  perfect  transparency.  (To  avoid  circum- 
locution, transparency  is  defined  throughout  this  paper  in 
terms  of  the  amplitude  of  the  transmitted  light,  not  in  terms 
of  intensity.)  Equation  (2)  is  evidently  the  development  of 
this  transparency  curve  as  a  series  of  harmonic  distributions 
of  transparency.  The  non-periodic  term  corresponds  to  a 
surface  of  uniform,  but  imperfect,  transparency  ;  while  each 
periodic  term  represents  a  surface  covered  with  equidistant 
parallel  bands  of  varying  transparency.  Supposing  these  bands 
to  run  perpendicular  to  the  plane  of  the  paper,  the  distribution 
of  transparency  in  them  wiil  be  indicated  by  the  ordinates 
in  fig.  2,  where  the  axis  of  x  again  represents  perfect  opacity, 
the  ordinate  B  a  certain  degree  of  transparency,  and  the 
ordinate  —  B  represents  an  equal  degree  of  what  we  may  term 
negative  transparency,  i.  e.,  transparency  coupled  with  a  half- 
period  change  of  phase  in  the  transmitted  light.  A  surface 
of  this   sort  may  be  called  a  simple  harmonic  grating  or, 
