124  Lord  Iiavleigh  on  the 
endeavour  to  define  more  clearly  the  position  which  I  am 
disposed  to  favour  on  one  or  two  o£  the  matters  concerned. 
On  p.  580,  in  comparing  white  light  and  Rontgen  radiation, 
Prof.  Larm  or  writes:  "  Both  kinds  of  disturbance  are  resolvable 
by  Courier's  principle  into  trains  of  simple  waves.  But  if 
wTe  consider  the  constituent  train  having  wave-length  variable 
between  X  and  \  +  o\,  i.e.  varying  irregularly  from  part  to 
part  of  the  train  within  these  limits,  a  difference  exists 
between  the  two  cases.  In  the  case  of  the  white  light  the 
vibration-curve  of  this  approximately  simple  train  is  in 
appearance  steady  ;  it  is  a  curve  of  practically  constant 
amplitude,  but  of  wave-length  slightly  erratic  within  the 
limits  SX  and  therefore  of  phase  at  each  point  entirely  erratic. 
In  the  Fourier  analysis  of  the  Rontgen  radiation  the  ampli- 
tude is  not  regular,  but  on  the  contrary  may  be  as  erratic  as 
the  phase/"  This  raises  the  question  as  to  the  general 
character  of  the  resultant  of  a  large  number  of  simple  trains  of 
approximately  equal  wave-length.  In  what  manner  will  the 
resultant  amplitude  and  phase  vary?  In  several  papers* 
I  have  considered  particular  cases  of  approximately  simple 
waves,  showing  how  they  may  be  resolved  into  absolutely 
simple  trains  of  approximately  equal  wave-lengths.  But  now 
the  question  presents  itself  in  the  converse  form.  TVnat  are 
we  to  expect  from  the  composition  of  simple  trains,  severally 
represented  by 
ax cos  {(n  +  BnJt  +  ei} (1), 
where  Snx  is  small,  while  the  amplitude  ay  and  the  initial 
phase  e°!  vary  from  one  train  to  another  ? 
In  virtue  of  the  smallncss  of  hiy  we  may  appropriately 
regard  (1)  as  a  vibration  of  speed  n  and  of  phase  €i-\-hnlt^ 
variable  therefore  with  the  time.  The  amplitude  and 
phase  may  be  represented  in  the  usual  way  by  the  polar 
coordinates  of  a  point  ;  and  the  point  representing  (1) 
accordingly  lies  on  the  circle  of  radius  ax  and  revolves 
uniformly  with  small  angular  velocity.  For  the  present  at 
any  rate  1  suppose  that  the  amplitudes  ax,  a2,  &c.  are  all 
equal  (1),  in  which  case  the  points  lie  all  upon  the  same  circle. 
The  radius  from  the  centre  0  to  any  of  the  points  P  upon 
the  circumference  is  a  vector  fully  representative  of  the 
vibration,  and  the  resultant  of  the  vectors  represents  the 
resultant  of  the  vibrations. 
After  the  lapse  of  a  time  t  the  points  have  moved  from 
their  initial  positions  P  to  other  positions  Q,  and  the  aggre- 
gate of  the  vectors  OP  is  replaced  by  the  aggregate  of  OQ. 
*  See  especially  Phil.  Mag.  vol.  l.p.  135  (1900). 
