This  is  only  another  way  of  expressing  the  fact  that  the  various
orders  of  spectra  are  superposed,  each  superposed  wave-length


6


being  inversely  proportional  to  its  order.  This  has  already
been  proved  before.  Furthermore,  the  equation  shows  that  the
dispersion  is  a  minimum  when  p  =  o,  i.e.  at  the  centre  of
curvature  of  the  grating ;  the  linear  dispersion  is  then  equal  to


i86  SPECTROSCOPY

-J- ,  and  is  constant ;  that  is  to  say,  the  spectrum  is  normal  at

this  point.  It  is  possible  to  give  /  quite  a  large  value  and  still
find  the  spectrum  normal,  and  it  has  been  proved  ^  that  the  error

in  normality  is  equal  to    ^   , ;   the   error   is   therefore   pro-

portional  to  the  cube  of  the  linear  distance  from  the  centre  of
the  photographic  plate.  An  example  may  be  given  of  this.
In  the  case  of  the  largest  Rowland  concave  grating  at  present
in  the  market,  l>  =  o'oooi27  cm.  and  p  —  650 cm.,  and  putting
the  maximum  allowable  error  at  j^  Angstrom  unit,  then  we
have  (u  =  i) —

3  _  I  X  10^"  X  24  X  (650)^

0'OOOI27

and/  =  8  cm.  nearly.
It  is,  therefore,  possible  to  photograph  1 6  cm.  of  the  first
order  spectrum  (8  cm.  on  each  side  of  the  centre)  perfectly
normal  within  this  error,  which  is,  in  fact,  the  greatest
attainable  accuracy  with  our  present  measuring  apparatus.
Similarly,  it  can  be  found  that  with  the  same  grating  the
deviation  from  normality  at  the  ends  of  a  50  cm.  photograph
is  only  o'03  A.U.  in  the  first  order.  The  distances  are  closely
proportional  to  these  with  the  smaller  gratings,  as  may  readily
be  found  from  the  above  expression.
