692  Lord  Rayleigh  :  Some  Measurements  of 
plates  at  two  different  distances.  In  Fabry  and  Perot's  work 
the  sliding  interferometer  was  employed  ;  the  silvered  surfaces 
were  brought  to  very  small  distances,  and  the  coincidences  of 
two  band  systems,  e.  g.  cadmium  red  and  cadmium  green,  were 
observed,  the  telescope  being  focussed  upon  the  plate,  and  not 
as  before  for  infinity.  It  appears  that  excellent  results  were 
obtained  in  this  way,  affording  material  for  eliminating  the 
complication  due  to  change  of  optical  thickness. 
It  is  rather  simpler,  in  principle,  and  has  the  incidental 
advantage  of  allowing  the  sliding  interferometer  to  be  dis- 
pensed with,  if  we  follow  the  same  method  for  the  small  as 
for  the  greater  distance.  If  the  calculation  be  conducted  on 
the  same  lines  as  before  by  means  of  (5),  we  ought  to  obtain 
the  same  fractional  part  again  in  the  value  of  P',  e.  g.  '95  for 
cadmium  green  referred  to  cadmium  red.  For,  as  we  see 
from  (6),  the  proportional  error  in  P'/P  as  calculated  from  (5) 
is  {e\'  —  ek)jek.  In  the  second  set  of  operations,  writing  tj  for  e, 
we  find  as  the  proportional  error  (yy  —  y^lvx,  in  which 
V\'  —  V\  =  e\>  ~ex  ;  so  that  the  proportional  errors  are  as  rjx  :  eXy 
or  inversely  as  P  or  P'.  Thus  the  absolute  error  in  IJ/,  as 
calculated  from  (5),  is  unaffected  by  the  change  of  e  to  77. 
If  the  fractional  part  is  not  recovered, within  the  limits  of  error, 
it  is  a  proof  that  the  assumed  ratio  of  wave-lengths  calls  for 
correction,  and  the  discrepancy  gives  the  means  for  effecting 
such  correction. 
The  above  procedureis  the  natural  one,  when  it  is  a  question  of 
identifying  a  ring  or  of  confirming  ratios  of  wave-lengths 
already  presumably  determined  with  full  accuracy  ;  but  when 
the  object  is  to  find  more  accurately  wave-lengths  only  roughly 
known,  it  has  an  air  of  indirectness.  Otherwise,  we  have 
as  before, 
2e}=p\         2e*=p'\'; 
and  again  for  a  smaller  interval  between  the  surfaces, 
2Vk  =  tt\  2Vk  =tt'V. 
Hence 
Ke\— Vk)  =  (P— "■)*■«       2(eK,-—w)={p'  —  ir')\'  ; 
an  d  eK  —  nx  —  ek —  rjx,     so  that 
-\  I  t 
w=y 7; 
A/  p  —  TT 
Hence  p,  it,  p\  it'  are  the  ordinal  numbers  at  the  centre. 
They  are  to  be  deduced,  as  before,  from  the  integral  numbers 
proper  to  the  rings  actually  observed  and  from  the  measured 
angular  diameters  of  these  rings. 
