Formula  (12)  applies  only  approximately,  but  it  is  always  found
that  the  frequencies  of  the  spectral  lines  can  be  written,  as  in
formulae  (2)  and  (12),  as  a  difference  of  two  functions  of  integral
numbers.  Thus  the  latter  formula  applies  accurately,  if  the
quantities  ak  are  not  considered  as  constants,  but  as  representatives
of  a  set  of  series  of  numbers  a&  (n)  characteristic  of  the  element,
whose  values  for  increasing  n  within  each  series  quickly  approach


PHYSICAL  AND  CHEMICAL  PROPERTIES  OF  THE  ELEMENTS       77

a  constant  limiting  value.  The  fact  that  the  frequencies  of  the
spectra  always  appear  as  the  difference  of  two  terms,  the  so-called
"spectral  terms,"  from  the  combinations  of  which  the  complete
spectrum  is  formed,  has  been  pointed  out  by  Ritz,  who  with  the
establishment  of  the  combination  principle  has  greatly  advanced
the  study  of  the  spectra.  The  quantum  theory  offers  an  immediate
interpretation  of  this  principle,  since,  according  to  the  frequency
relation  we  are  led  to  consider  the  lines  as  due  to  transitions
between  stationary  states  of  the  atom,  just  as  in  the  hydrogen
spectrum,  only  in  the  spectra  of  the  other  elements  we  have  to  do
not  with  a  single  series  of  stationary  states,  but  with  a  set  of  such
series.  From  formula  (12)  we  thus  obtain  for  an  arc  spectrum — if
we  temporarily  disregard  the  structure  of  the  individual  lines —
information  about  an  ensemble  of  stationary  states,  for  which  the
energy  of  the  atom  in  the  nth  state  of  the  kth  series  is  given  by

*<">— <s£? (13)

very  similar  to  the  simple  formula  (3)  for  the  energy  in  the  station-
ary states  of  the  hydrogen  atom.
