Diffraction  Theory  of  Microscopic  Vision. 
157 
briefly,  a  simple  grating.     A  simple  grating  would  give  spectra 
of  the  first  order  only,  and  no  central  image.     Such  a  grating 
Fio-.  2. 
could  perhaps  be  realized  by  covering  the  alternate  transparent 
bands  on  a  harmonically  shaded  plate  with  strips  of  a  trans- 
parent film  giving  a  half  wave-length  retardation. 
4.  It  now  follows  immediately  from  equation  (2)  that  the 
grating  of  fig.  1,  and,  indeed,  in  the  more  general  case,  any 
opacity  grating  ruled  with  n  lines  per  unit  length,  may  be 
considered  to  be  formed  by  the  superposition,  on  an  imper- 
fectly transparent  surface,  of  a  series  of  simple  gratings  ruled 
with  n,  2n,  3n,  4rz,  &c.  lines  per  unit  length.  The  imperfectly 
transparent  surface  is  responsible  for  the  central  image,  while 
each  of  the  simple  gratings  gives  rise  to  the  spectra  of  one 
order,  the  amplitude  of  the  light  in  each  spectrum  being 
proportional  to  the  amplitude  of  the  transparency  curve  of 
the  corresponding  simple  grating.  Furthermore,  it  is  easily 
seen  that  the  necessary  and  sufficient  condition  for  the  absence 
of  the  spectra  of  any  given  order  is  the  absence  of  the  corre- 
sponding simple  grating. 
5.  The  theory  here  briefly  indicated  may  be  extended  to 
include  retardation  gratings,  both  transparent  and  reflecting, 
but  this  development  is  beyond  the  scope  of  this  paper.  One 
deduction  from  the  theory  may,  however,  be  mentioned 
because  of  the  ease  with  which  it  can  be  verified  experimentally. 
The  sharply  ruled  grating  of  fig.  1  consists  of  an  infinite 
series  of  simple  gratings  which  give  a  fan  of  spectra  on  each 
side  of  the  central  image.  If  the  sharp  corners  in  fig.  1  are 
rounded,  that  is  to  say  if  the  lines  of  the  grating  are  blurred, 
Fourier's  theorem  indicates  the  absence  of  the  higher 
harmonic  terms  in  equation  (2),  and  hence  also  the  absence 
of  the  more  finely  spaced  simple  gratings  and  the  corresponding 
spectra  of  higher  orders.     When  the  lines  are  much  binned. 
