Since  the  sums  of  the  values  of  p,  corresponding  to  positive  and  negative
values  of  q,  separately  vanish  (and  vice  versa)  the  summation  {pq)  will  be
small  compared  with  the  others,  and  we  shall  have

the  actual  values  usually  approaching  closely  to  equality.  Now,  unless
all  the  />'s  are  numerically  equal  {p')>nP'.     We  may  set  {p')=KnP',

where  K>i.     If  the  stars  are  uniformly  distributed  over  the  plate,  K  =  -.
They  can  hardly  in  practice  be  so  irregularly  grouped  that  K  is  greater  than  2.
We  then  have  r^^ :=^,  whence  by  (9)  r^  ^  rj^K,  and  similarly  r„^  r,  i^ K.
The  diminution  in  the  probable  errors  of  these  plate-constants,  upon
passing  from  the  approximate  method  to  that  of  least-squares,  is  therefore
usually  small.
For  the  actual  difference  between  the  two  values  of  a,  we  have,  in  the
case  where  ipq)=o

.4     Kn-^^^^      2n,(A),      2n,(A).  ,.
The  most  probable  value  of  this  difference  is  obviously  zero.  Its  probable
error  R^  may  be  obtained  by  squaring  the  second  member  of  (12),  substitut-
ing r'  for  the  square  of  each  of  the  A'^,  and  zero  for  all  product  terms.  We
thus  obtain

p,_    r'     ,  4(nln,+nln,y      4n.iP).r'      4n,(P)r
^     (/>0  n*P'  n'Pip')  "^  n'Pip')

By  means  of  (7),  (8),  and  (11)  this  reduces  to

Rl  =  rl+rl-  ^^'';+r^-^''  =rl-rl^rl{K-i)  (13)

When  (pq)  is  not  zero,  the  same  final  equation  is  reached,  after  a  slightly
longer  reckoning.
