These  show  a  distinctly  systematic  variation  with  the  magnitude,  the
only  exception  being  the  last  group  but  one.  The  mean  for  this  is  raised  by
the  presence  of  one  very  bad  star  (No.  33).  The  original  observations  show
that  the  abnormally  large  probable  error  for  this  star  is  almost  entirely  due
to  the  extreme  discordance  of  a  single  plate,  on  which  its  images  were  recorded
at  the  time  of  measurement  as  excessively  faint.  It  is  questionable  whether
this  plate  should  have  been  measured  at  all.  In  any  case,  it  does  not  rep-
resent the  normal  probable  error  for  stars  of  this  magnitude.  A  second  mean
has  therefore  been  formed  for  this  group,  excluding  this  star.  The  legitimacy
of  this  process  is  confirmed  by  the  fact  that  if  the  under-exposed  plate  were
rejected  the  probable  error  derived  for  this  star  from  the  remaining  plates
would  have  been  close  to  the  mean  for  the  rest  of  the  group.  The  means,
after  this  correction,  can  be  closely  represented  by  the  formula

r='  =  io-|-4  (m-9)'  (i)

where  r  is  the  probable  error,  expressed  in  hundredths  of  a  second  of  arc,  and
m  the  photographic  magnitude — as  is  shown  by  table  30.
The  average  duration  of  exposure  for
the  different  groups  is  so  nearly  equal  that
the  extreme  magnitudes  need  be  changed  by
little  more  than  one-tenth  of  a  unit  to  reduce
the  results  for  all  the  groups  to  the  average
exposure  of  4"3 .  The  corrections  to  the  effec-
tive photographic  brightness  of  the  stars  on
different  plates,  depending  on  the  clearness
of  the  air,  etc.,  would  doubtless  be  much
larger  than  this.  The  above  formula  may
therefore  be  applied  with  sufficient  approximation  to  all  the  plates.
