in

For  another  series  we  have —

00
n'„  =  «'o  —  2i  7{m  +  /*')•

Now,  the  two  sums  which  enter  into  these  equations  have  the


SERIES   OF  LINES   IN  SPECTRA  567

If  we  put_/(/«  +  /i)=  — — —  we  have  the  equation —


n  =  n.  —


m  +  fjL
We  have  already  seen  that  a  very  similar  equation —

X  =  X„  +  S^

m  -\-  fj.
expresses  a  series  with  a  very  rough  approximation.  Con-
sidering the  shape  of  the  curves  already  described,  obtained  by
plotting  X  against  m,  the  next  simplest  equation  is —

No

where  Nj  is  another  constant.
This  equation  Rydberg  finds  to  answer  extraordinarily  well,
and  is  the  equation  he  finally  adopts ;  in  it  No  is  a  constant  for
all  series  and  all  substances,  «,  is  the  value  of  n  when  m  =  00  ;
that  is  to  say,  it  is  the  theoretical  limit  of  the  series,  and  is
generally  known  as  the  convergence  frequency,  and  /a  is  a
characteristic  constant  of  the  substance  or  series.
Now,  Balmer's  hydrogen  equation  may  be  considered  as
a  special  case  of  this  formula,  for  if  instead  of  his  equation  for
the  wave-length —


wz^  —  4


»r  —  4
we  put  ft  =  '^o—^ — ,


10°  10

when  n  =  ^—  and  ??»  =


X'  ^"d  '^«  =  ^


same  number  of  terms,  and  we  know  that  any  term  in  one  of  the
sums  only  differs  from  the  corresponding  term  in  the  other  by
the  value  of  the  constant  fi ;  then  one  of  the  sums  is  changed  into
the  other  if  n  is  changed  to  /i,  whence  it  follows  that  there  are
only  two  different  values  of  the  same  function  of  li.
