Resolving  Power  of  Spectroscopes.  415 
depends  only  on  the  total  number  of  lines  on  the  grating 
which  are  utilized.  In  practice,  however,  we  frequently 
make  use  of  slits  so  wide  that  the  theoretical  resolving  power 
is  far  from  realized.  This  case  was  considered  by  Schuster, 
who  introduced  the  idea  of  u  purity  "  of  spectra. 
If  light  containing  two  homogeneous  radiations  differing 
in  wave-length  by  h\  falls  on  the  slit  of  a  spectroscope,  the 
condition  for  resolution  of  the  corresponding  lines  has  been 
shown  by  Schuster  to  be 
h\_  2\-\-yfrd 
\~      \R     ? 
where  d  is  the  width  of  the  slit,  R  the  theoretical  resolving 
power  as  defined  by  Rayleigh,  and  i/r  is  the  angle  subtended 
at  the  slit  by  the  diameter  of  the  coliimator-lens. 
This  expression  appears  to  contain  no  factor  depending  on 
the  position  of  the  grating  or  prism  ;  but  a  simple  consider- 
ation shows  that  the  purity  becomes  greater  as  the  angle  of 
incidence  of  the  light  on  the  grating  increases. 
When  the  slit  has  a  finite  width  we  can  consider  the  light 
falling  on  the  grating  to  consist  of  a  number  of  beams  of 
parallel  rays  making  slightly  different  angles  with  the  normal 
to  the  grating. 
Let  (/>  be  angle  of  incidence  and  6  the  angle  which  the  cor- 
responding diffracted  beam  makes  with  the  normal. 
lie  is  the  grating  space,  we  have  the  usual  relation 
e(sin  (f>  —  sin  &)  =  +  n — ; 
from  which  ,^  __  cos  cj) 
and  dO 
COS  V 
dX      2e cos  6 
for  any  value  of  (p. 
The  diffracted  beams  will  thus  make  slightly  different 
angles  with  the  normal,  the  total  variation  being  dQ.  This 
variation  can  be  made  as  small  as  we  please  by  increasing 
the  value  of  </>  so  that  in  the  limit  when  the  light  falls  on  the 
grating  at  grazing  incidence  the  diffracted  beams  corre- 
sponding to  any  particular  wave-length  and  order  all  make 
the  same  angle  with  the  normal,  and  will  be  received  in  the 
telescope    under  nearly  the  same  conditions  as  if  the  colli- 
mator-slit    had    been    verv    narrow.       Further,    -—    is    also 
dX 
increased,  so  that  the  total  effect  is  to  greatly  increase   the 
purity  of  the  spectrum.    . 
