162  Mr.  A.  B.  Porter  on  the 
ij.  Certain  appearances  noted  in  the  image  of  a  grating  pro- 
duced by  a  microscopic  objective  stopped  down  as  described 
in  paragraph  6  suggested  the  advisability  of  a  closer  study 
of  these  effects.  The  method  outlined  in  paragraph  2  is 
sufficient  to  determine  the  nature  of  the  image  of  any  opacity 
grating  given  by  a  lens  of  any  aperture.  The  curve  showing 
the  distribution  of  amplitude  in  the  plane  of  the  grating  of  the 
light  which  passes  through  it  is  developed  in  a  Fourier  series, 
both  sines  and  cosines  being  used  if  the  lines  of  the  grating- 
are  shaded  unsymmetrically.  From  the  aperture  of  the  lens 
and  the  spacing  of  the  lines  of  the  grating  the  number  of 
orders  of  spectra  transmitted  by  the  lens  is  determined  and, 
remembering  that  each  periodic  term  in  the  Fourier  series 
corresponds  to  the  spectra  of  one  given  order,  all  terms  in  the 
series  are  rejected  which  represent  spectra  not  transmitted 
by  the  lens.  The  remaining  terms  when  summed  give  the 
amplitude  curve,  in  the  plane  of  the  grating,  of  the  effective 
light  which  is  transmitted  by  the  lens.  The  amplitude  curve 
in  the  image  differs  from  this  merely  in  having  its  length 
increased  and  its  height  diminished  in  proportion  to  the  mag- 
nification. By  squaring  the  ordinates  of  this  curve,  the 
intensity  curve  is  obtained  which  completely  determines  the 
structure  of  the  image.  Intensity  curves  were  roughly  drawn 
to  scale  in  this  way  for  the  images  of  several  different  gratings 
and  for  a  number  of  different  apertures.  Figs.  6-1  to  6-XI 
show  a  set  of  these  intensity  curves  giving  the  distribution  of 
light  in  the  images,  formed  by  lenses  of  different  apertures, 
of  a  grating  whose  opaque  lines  are  twice  as  broad  as  the 
transparent  spaces.  In  each  case  the  distribution  of  light  in 
two  bright  lines  is  shown,  the  base  line  of  each  figure 
representing  darkness.  The  Roman  numeral  indicates  in 
each  case  the  highest  order  of  spectra  transmitted  by  the 
lens.  Fig.  6-1  shows  the  image  formed  by  a  lens  of  aperture 
just  sufficient  to  transmit  the  spectra  of  the  first  order,  and 
indicates  the  presence  of  a  moderately  bright  streak  a  down 
the  middle  of  each  dark  line  in  the  image.  Fig.  6-II  shows 
the  image  formed  by  a  lens  transmitting  the  spectra  of  the 
first  two  orders  ;  the  image  is  much  sharper  than  that  of 
fig.  6-1.  The  third,  sixth,  ninth,  &c,  orders  are  absent  with 
the  grating  under  consideration.  Figs.  6-IV  and  6-V  show 
the  nature  of  the  images  when  the  aperture  is  further  widened 
go  that  spectra  up  to  and  including  the  fourth  and  fifth  orders 
respectively  are  transmitted  by  the  lens.  The  noteworthy 
point  here  is  the  appearance  of  a  dark  streak  down  the  centre 
of  each  bright  line.  This  is  particularly  interesting  because 
it  is  a  defect,  or  rather  a  falsification,  in  the  image  which 
