164  Mr.  A.  P.  Porter  on  the 
appeared  down  the  centre  .of  each  (fig.  6-V)  which  faded 
away  as  the  iris  was  further  opened,  to  be  succeeded  by  two 
dark  streaks  (fig.  6-VII),  then  three  fainter  ones  (fig.  6-XI) 
which  soon  disappeared  with  further  enlargement  of  the 
aperture.  When  the  grating  was  illuminated  by  white  light 
the  bright  streaks  shown  in  fig.  6-1  could  be  seen  but,  on 
account  of  the  overlapping  of  the  spectra  of  the  higher  orders, 
the  dark  streaks  indicated  in  figs.  6-IV  to  6-XI  did  not 
appear  in  the  image.  These  experiments  suggest  that  some 
caution  should  be  used  in  interpreting  minute  details  of 
structure  in  microscopic  images  when  monochromatic  illu- 
mination is  used. 
10.  The  curves  given  in  figs.  6-1  to  6-XI  show  the 
intensity  of  the  light  in  all  parts  of  the  image.  They  are 
also  interesting  from  another  point  of  view  for,  from  what 
has  been  said  in  paragraphs  4  and  5,  it  will  be  seen  that 
each  of  these  curves  shows  the  exact  distribution  of  trans- 
parency (defined  now  in  terms  of  the  intensity  of  the 
transmitted  light)  over  the  surface  of  a  grating  which 
gives  spectra  of  certain  definite  orders  and  definite  intensities. 
Thus  a  grating  having  lines  shaded  as  indicated  in  fig.  6-IV 
w^ould  give  a  central  image  and  spectra  of  the  first,  second, 
and  fourth  orders  only,  and  these  spectra  wTould  have  exactly 
the  same  intensities  as  those  given  by  a  sharply  ruled  grating 
whose  opaque  lines  are  twice  as  broad  as  the  transparent  spaces. 
This  application  of  Fourier's  theorem  enables  one,  in  fact. 
to  design  an  opacity  grating  presenting  any  desired  anomaly. 
The  realization  of  such  gratings  is  quite  a  different  matter, 
but  it  is  hoped  that  some  results  may  be  presented  in  a  Liter 
paper. 
11.  Transparent  objects  Avhich  are  subjected  to  microscopic 
examination  are  ahvays  of  finite  thickness,  and  it  is  of  some 
interest  to  inquire  whether  light  diffracted  by  portions  of 
the  object  lying  outside  the  focal  plane  may  not  modify  the 
image.  The  mathematical  analysis  of  this  question  promising 
to  be  troublesome  and  unprofitable,  an  experimental  illustra- 
tion of  the  suspected  effect  was  sought  and  one  was  finally 
found  which  is  extremely  striking.  It  is  well  known  that 
when  two  transmission  gratings  are  laid  face  to  face  with 
lines  parallel  and  ruled  surfaces  slightly  separated,  the  spectra 
formed  by  the  double  grating  are  intersected  by  transverse 
shadow-bands  whose  spacing  depends  upon  the  distance 
between  the  faces  of  the  grating.  Tw^o  photographic  gratings 
with  3000  lines  to  the  inch  thus  arranged  with  faces  about 
a  millimetre  apart,  showed  a  black  band  in  each  of  the  twro 
spectra   of  the  first    order   and,  by  changing  the    angle   of 
