534  2Tote  on  Talbot's  Lines. 
<f>  be  the  angle  of  incidence,  the  disturbance  in  a  direction^ 
may  be  represented  by 
(9<7T  \  973- 
—  vt—.A)+A  cos  —  {vt  —  /i(sin  6  +  sin  <j>)  } 
=  2A  cos  J  ~h  (sin  0  +  sin  0)  -  f  J 
r2  tt  r  7t(sin0-f  sin<f>)  )        Al 
2A  being  the  width  of  the  aperture,  A  die  retardation  of  phase 
introduced  by  the  plate,  and  0,  <p  being  regarded  as  positive 
when  measured  on  the  side  of  the  normal  on  which  the  plate 
is  inserted. 
Hence  the  intensity  is 
4A2  cos2  {  ^-h  (sin  0  +  sin  <j>)-^\, 
and  there  will  be  a  primary  series  of  minima  given  by  A  =  0, 
being  those  due  to  a  rectangular  aperture  of  width  h,  and  a 
secondary  series  of  minima  in  directions  given  by 
7*(sin0  +  sm<£)  =  \~±{2n-l)  }|,      n  =  l,  2,  .  .  . 
Let  the  telescope  be  directed  so  that  light  of  wave-length 
\0  is  incident  normally  upon  the  aperture,  and  let  p  be  the 
order  of  the  secondary  minimum  next  to  the  focal  point. 
Then  0  being  very  small, 
he 
-{i-*-")}* 
and  for  light  of  wave-length  \0  +  S\0 
A(0  +  S0  +  8*)=  {--(2p-l)  IH^  +  ^SX 
■whence  [w     v  1        '  1  2        iirdx^     ' 
v  y       27rdX0 
Hence  for  coincidence  of  the  bands  due  to  the  monochromatic 
constituents  \0  and  X0  +  8\,  we  must  have 
dA  _  2irli  d(f> 
which  gives  the  best  thickness  of  the  plate. 
X1      dA      A  af(j)  ,  .  tf  A 
Also  — —  and  —-  must  nave    the    same   sign :  whence  ^,— 
d\  dX  °  rfX, 
being  negative.,  the  angle  of  incidence  must  increase  on  the 
positive  side  of  the  normal  with  decreasing  wave-length. 
